# Cooperative Guidance Law with Predefined-Time Convergence for Multimissile Systems

Authors: Liang Jing; Changzhu Wei; Liang Zhang; Naigang Cui
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

To intercept the maneuvering target through multimissile cooperation, predefined-time cooperative guidance (PTCG) law is presented with constraints including the impact time and the line-of-sight (LOS) angle. In order to achieve simultaneous interception, we propose a PTCG law in the LOS direction based on a predefined-time consensus protocol, which guarantees the achievement of consensus on each missile’s impact time within the predefined time. Furthermore, to ensure the predefined-time convergence of the LOS angle and the predefined-time convergence of the LOS angular rate, a PTCG law with a fixed-time disturbance observer (FxTDO) is presented in the normal direction of the LOS. Compared with the traditional finite-time or fixed-time cooperative guidance laws, the proposed PTCG law predefines the upper bound of the settling time as an explicit parameter. Finally, the simulation results of the PTCG law verify the efficiency of the proposed method.


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## Body

## 1. Introduction

In recent years, with the continuous development of modern military technology, the maneuverability of the target has been remarkably improved, and thus the single-missile combat mode has become more difficult. To overcome this problem, the multimissile cooperative interception is put forward, which has received extensive attention [1, 2].The cooperative interception guidance law requires stringent impact time constraints for multiple missiles to achieve simultaneous arrival. Driven by the requirement, two methods are adopted to complete the cooperative interception mission: the individual homing guidance law [1–14] with the impact time constraint and the cooperative guidance law [15–20]. The individual homing guidance law does not consider communication among missiles. To simultaneously reach the target position, Jeon [4] first proposed the impact time control guidance law (ITCG), which uses the error feedback of the impact time to adjust the coefficient of proportional navigation guidance. Then, a new ITCG was designed in [6], which combines the impact time error feedback with the sliding mode control. Nevertheless, in the aforementioned guidance laws, the impact time is predefined and each missile hits the target according to its own guidance law. Hence, the impact time is not adjustable in flight.By contrast, the cooperative guidance law deals with the problem of cooperative interception involving communication among multiple missiles. In [15], based on the leader-follower scheme, a leader-oriented cooperative guidance law was proposed. The leader can coordinate the impact time of followers by the cooperative guidance law. But this method requires high real-time quality and high reliability for the communication of the leader. To tackle this problem, the distributed cooperative guidance law was presented in [16–20], which can relax the topological communication requirement and has stronger robustness. To improve the information transmission capacity, consensus protocols for multiagent systems [21–23] were introduced into the design of distributed cooperative guidance. By utilizing the consensus protocol algorithm, agents can reach a consensus based on local information. To guarantee the multiple missiles achieve a consensus at a faster convergence rate in the cooperative interception missions, the finite-time consensus was put forward. In [24], Lv presented a distributed finite-time cooperative guidance (FTCG) law with angle constraints, which utilizes the finite-time consensus to improve the convergence rate and robustness. Then, the FTCG law with an acceleration saturation constraint was proposed in [25], which is characterized by its fast and accurate convergence of the guidance system. Furthermore, since the acceleration of the target is unmeasurable during the flight, it could be considered as a disturbance. And the influence of the unmeasurable disturbance can be reduced by the disturbance observer. In [26], Guo and Liang presented a nonsingular fast terminal sliding mode cooperative guidance algorithm without the target maneuver information, where the unknown target acceleration is estimated by a finite-time disturbance observer. Moreover, a new FTCG law based on the finite-time consensus protocol with LOS angle constraints was presented in [27], which uses an extended state disturbance observer. Then, the FTCG law [27] was introduced into a three-dimensional (3D) multimissile guidance system in [28]. However, for the disturbance observers and guidance laws in [24–29], their convergence time is affected by initial conditions and increases unboundedly as the initial errors increase, which dramatically affects the accuracy of the system.Fixed-time control was put forward and gained increasing attention following finite-time control. For a fixed-time stable system, a constant independent of initial conditions can uniformly bound the system’s convergence time. In the existing research literature, fixed-time control schemes were adopted to design the cooperative guidance law because fixed-time stability enjoys fast convergence and strong robustness. In [30], a distributed fixed-time cooperative guidance (FxTCG) law with an FxTDO was proposed, which guarantees that each missile can reach a consensus within the fixed time. Subsequently, Zhang and Ma [31] developed an adaptive FxTCG law with LOS angle constraints. However, the negative power term exists in the consensus protocol in [31]. To solve this problem, a new FxTCG law was presented in [32], which utilizes an adaptive fixed-time convergence reaching law together with a novel fixed-time consensus protocol. Moreover, Chen et al. [33] proposed a 3D FxTCG law with LOS angle constraints for maneuvering target interception, which can achieve faster convergence and high precision. Lin et al. [34] presented a 3D FxTCG law with impact angle constraints for the simultaneous arrival problem, which can set the desired impact time in advance. However, the selection of fixed convergence time is not arbitrary, and there is usually a complex relationship between the tunable parameters and the fixed convergence time.To overcome the above drawbacks, Sanchez-Torres originally proposed the predefined-time stability theory in [35]. Predefined-time stability has an obvious advantage in the system settling time. The settling time’s upper limit can be arbitrarily set by choosing appropriate controller parameters. Predefined-time sliding mode controllers for dynamic systems were designed in [36–38]. Subsequently, the predefined-time control was used for controlling the attitude of rigid spacecraft and tracking robotic manipulators in [39, 40]. Besides, a novel predefined-time consensus protocol was proposed in [41], which ensures a predefined-time consensus for static topologies.Motivated by the mentioned above, this paper is dedicated to deducing a PTCG law with the LOS angle constraint as well as the impact time constraint. The main contributions are summarized below:(1)
A novel formulation of the predefined-time convergence is proposed. The parameter setting of the given predefined-time convergence is more flexible and convenient, and the limit of the settling time is treated as an explicit tuning parameter.(2)
A new predefined-time consensus protocol is presented with strict proof. And in the LOS direction, according to the above protocol, a PTCG law is proposed for multimissile intercepting the target simultaneously on the basis of predefined-time convergence as well as the given algebraic graph.(3)
In the normal direction of LOS, a PTCG sliding mode control law with an FxTDO is presented to ensure that the LOS angle and the LOS angle rate can realize predefined-time convergence. Considering the uncertain disturbance, an FxTDO is used to estimate the disturbance and compensate for the PTCG law.The remainder of this paper is structured as follows: Section2 introduces the cooperative guidance model and preliminary lemmas. A PTCG law is presented with strict proof in Section 3. The numerical simulation results and analysis are given in Section 4. Section 5 concludes this paper.


## 2. Problem Formulation and Preliminaries

### 2.1. Problem Formulation

Consideringn missiles intercept a maneuvering target simultaneously. Figure 1 shows a 2D planar engagement. The subscript i,i=1,2,…,n denotes the i-th missile. Mi and T indicate the missiles and target, respectively. XOY is the inertial frame.Figure 1
Planar guidance geometry.The relative kinematic model is given by(1)r˙i=VTcosqi−θT−VMicosqi−θMi,(2)riq˙i=−VTsinqi−θT+VMisinqi−θMi,(3)θ˙Mi=aMiVMi,θ˙T=aTVT,where VMi and VT indicate the velocity of the missiles and the target, respectively. ri is the distance from the missile to the target. θMi and θT denote the flight path angles of the missile and the target, respectively. The LOS angle is denoted as qi. aMi and aT represent the i-th missile’s and the target’s normal accelerations.The derivatives of equations (1) and (2) can be obtained as(4)r¨i=riq˙i2+uri−wri,q¨i=−2r˙iq˙iri−uqiri+wqiri,where uri and uqi are the i-th missile’s acceleration components along the LOS direction and the normal direction of the LOS. wri and wqi are the acceleration components of the target along the LOS direction and the normal direction of the LOS. In the actual interception mission, wri can be considered as zero.Definex1i=ri, x2i=r˙i, x3i=qi−qfi, and x4i=x˙3i=q˙i. Then, the guidance system is given as(5)x˙1i=x2i,x˙2i=x1ix4i2−uri,x˙3i=x4i,x˙4i=−2x2ix1ix4i−uqix1i+wqix1i,where(6)tgi=−rir˙i.Differentiating equation (6) yields(7)t˙gi=−1+x1i2x4i2x2i2−x1ix2i2uri.Considertgi as a new state variable, and the dynamic equations are rewritten as(8)t˙gi=−1+x1i2x4i2x2i2−x1ix2i2uri,x˙3i=x4i,x˙4i=−2x2ix1ix4i−uqix1i+wqix1i.Defineu˜ri as a new variable(9)u˜ri=x1i2x4i2x2i2−x1ix2i2uri.Substitute equation (9) into equation (8), and we have(10)t˙gi=−1+u˜ri,x˙3i=x4i,x˙4i=−2x2ix1ix4i−uqix1i+wqix1i.The research objectives of this paper are shown below. In the LOS direction, we designuri to make sure that tgi can reach a consensus within a predefined time. uqi is proposed to ensure that x3i and x4i can approach zero within a predefined time.


### 2.2. Graph Theory

Assumen missiles participate in the interception mission cooperatively. GN,I,W is the graph of information communication among the multiple missiles. GN,I,W is formed by node N=N1,N2,…,Nn, I⊆Ni,Nj:Ni,Nj∈N, and a weighted adjacency matrix W=cij∈Rn×n. When the information is transferred from Nj to Ni, the weight of edge cij satisfies cij≠0. Otherwise, cij=0. It is worth noting that G is undigraph if Ni,Nj∈I⇔Nj,Ni∈I. The adjacency matrix W has symmetry in the undigraph G. If the undigraph G is connected, there is a path connecting every two nodes [27]. The neighbor set of Ni can be obtained as(11)Mi=j∈N:cij≠0=j∈N:Ni,Nj∈I


### 2.3. Preliminaries

Definition 1 (see [40–42]).
Consider a nonlinear system:(12)x˙t=fxt,ϑ,x0=x0,where xt∈Rn denotes the system state variable and ϑ∈Rl is the design parameter of system (12). System (12) is said to converge to the origin within a finite time if there exists a settling time function T:Rn⟶R+, such that, for every x0∈Rn, the solution of system (12) satisfies limt⟶Tx0Ψt,x0=0. For a globally finite-time convergent system (12), if for any initial state x0∈Rn, there exists a bounded convergence time Tmax irrelevant to initial conditions satisfying Tx0<Tmax, the origin has global fixed-time convergence.Definition 2.
(see [42–44]). The origin of the fixed-time stable system (12) is predefined-time stable if, for any convergence time Tc∈R+, there is ϑ∈Rl such that the settling-time function of (12) satisfies Tx0x0∈ℝn≤Tc.Definition 3.
(see [38]). Define a continuous function κ:R+⟶0,1. It belongs to class-▿, and there is κ∈▿1. κ is monotonically increasing such that κ0=0 and limr⟶∞κr=1.
Notion 1 (see [38]). For a given vector x=x1,x2,…,xnT, it can be obtained that sigax=xasignx and sigax=x1asignx1,x2asignx2,…,xnasignxnT, where signx denotes the sign function.Lemma 1.
(see [38]). For system (12), define an unbounded Lyapunov function Vx and let κ∈▿1 be differentiable in R\0. For any predefined convergence time Tc∈R+ and ϑt,x0∈Rl, the time-derivative of Vx satisfies(13)V˙x≤−11−pTcκVPκ′V,forx∈Rn\0,where 0<p<1, such that system (12) is predefined-time stable.Lemma 2.
[45]. Define a Lyapunov function Vx and let κV=Vξ/Vξ+χ, with χ>0, such that(14)V˙≤−1ξχ1−pTcVξp−ξ+1Vξ+χ2−p+Ω,for any solution ϑt,x0 of system (12), where Tc>0 denotes a predeﬁned-time parameter, χ>0 is a tuning parameter to adjust the convergence speed, Ω>0 is a small positive number, and 0<ξ<1 stands for an exponential coefficient. Then, system (12) origin has global predeﬁned-time stability, with predeﬁned time Tc.Proof.
Firstly, system (14) can be transformed into a differential equation:(15)V˙=−1ξχ1−pTcVξp−ξ+1Vξ+χ2−p.
By integrating equation (15), there is(16)Tx0=−∫Vx00dV1/ξχ1−pTcVξp−ξ+1Vξ+ϒ2−p=ξχ1−pTc∫0Vx0dVVξp−ξ+1Vμ+χ2−p=1−pTc∫0Vx0ξχVξ−1/Vξ+χ2Vξ/Vξ+χpdV=TcVξVξ+χ1−p|0Vx0.=Tc⋅Vξx0Vξx0+χ1−p
It is obvious thatlimVx0⟶∞Vξx0/Vξx0+χ=1 such that limVx0⟶∞Tx0=Tc. Therefore, the upper bound of the settling time is Tc.Theorem 1.
Define a radially unbounded Lyapunov functionVx and let κV=2/πarctanV such that(17)V˙≤−2πp−11+V21−pTcarctanVp+Ω,for any solution ϑt,x0 of system (12), where Tc>0, 0<p<1, and Ω>0 is a small positive number. Then, the origin of system (12) has predeﬁned-time stability, with predeﬁned time Tc.Proof.
Firstly, system (17) can be transferred as a differential equation:(18)V˙=−2πp−11+V21−pTcarctanVp.
By integrating with equation (18), there is(19)Tx0=−∫Vx00dV1/1−pTcπ1+V22/πarctanVp/2=21−pTcπ∫0Vx0dV1+V22/πarctanVp=1−pTc∫0Vx02/π1+V22/πarctanVpdV=Tc2πarctanV1−p|0Vx0=Tc⋅2πarctanVx01−p.
It is obvious thatlimVx0⟶∞2/πarctanVx0=1 such that limVx0⟶∞Tx0=Tc. Therefore, the bound of the settling time is Tc. Then, the origin of system (12) has predeﬁned-time stability, with predeﬁned time Tc.Lemma 3 (see [46]).
For anyyi∈R,i=1,2,…,n∑i=1nyip≤∑i=1nyip, where p∈R+ and p∈0,1.Lemma 4 (see [46]).
Ifη∈R+ and η>1, for any x,y∈R, there exists x+yη≤2η−1xη+yη.


## 2.1. Problem Formulation

Consideringn missiles intercept a maneuvering target simultaneously. Figure 1 shows a 2D planar engagement. The subscript i,i=1,2,…,n denotes the i-th missile. Mi and T indicate the missiles and target, respectively. XOY is the inertial frame.Figure 1
Planar guidance geometry.The relative kinematic model is given by(1)r˙i=VTcosqi−θT−VMicosqi−θMi,(2)riq˙i=−VTsinqi−θT+VMisinqi−θMi,(3)θ˙Mi=aMiVMi,θ˙T=aTVT,where VMi and VT indicate the velocity of the missiles and the target, respectively. ri is the distance from the missile to the target. θMi and θT denote the flight path angles of the missile and the target, respectively. The LOS angle is denoted as qi. aMi and aT represent the i-th missile’s and the target’s normal accelerations.The derivatives of equations (1) and (2) can be obtained as(4)r¨i=riq˙i2+uri−wri,q¨i=−2r˙iq˙iri−uqiri+wqiri,where uri and uqi are the i-th missile’s acceleration components along the LOS direction and the normal direction of the LOS. wri and wqi are the acceleration components of the target along the LOS direction and the normal direction of the LOS. In the actual interception mission, wri can be considered as zero.Definex1i=ri, x2i=r˙i, x3i=qi−qfi, and x4i=x˙3i=q˙i. Then, the guidance system is given as(5)x˙1i=x2i,x˙2i=x1ix4i2−uri,x˙3i=x4i,x˙4i=−2x2ix1ix4i−uqix1i+wqix1i,where(6)tgi=−rir˙i.Differentiating equation (6) yields(7)t˙gi=−1+x1i2x4i2x2i2−x1ix2i2uri.Considertgi as a new state variable, and the dynamic equations are rewritten as(8)t˙gi=−1+x1i2x4i2x2i2−x1ix2i2uri,x˙3i=x4i,x˙4i=−2x2ix1ix4i−uqix1i+wqix1i.Defineu˜ri as a new variable(9)u˜ri=x1i2x4i2x2i2−x1ix2i2uri.Substitute equation (9) into equation (8), and we have(10)t˙gi=−1+u˜ri,x˙3i=x4i,x˙4i=−2x2ix1ix4i−uqix1i+wqix1i.The research objectives of this paper are shown below. In the LOS direction, we designuri to make sure that tgi can reach a consensus within a predefined time. uqi is proposed to ensure that x3i and x4i can approach zero within a predefined time.


## 2.2. Graph Theory

Assumen missiles participate in the interception mission cooperatively. GN,I,W is the graph of information communication among the multiple missiles. GN,I,W is formed by node N=N1,N2,…,Nn, I⊆Ni,Nj:Ni,Nj∈N, and a weighted adjacency matrix W=cij∈Rn×n. When the information is transferred from Nj to Ni, the weight of edge cij satisfies cij≠0. Otherwise, cij=0. It is worth noting that G is undigraph if Ni,Nj∈I⇔Nj,Ni∈I. The adjacency matrix W has symmetry in the undigraph G. If the undigraph G is connected, there is a path connecting every two nodes [27]. The neighbor set of Ni can be obtained as(11)Mi=j∈N:cij≠0=j∈N:Ni,Nj∈I


## 2.3. Preliminaries

Definition 1 (see [40–42]).
Consider a nonlinear system:(12)x˙t=fxt,ϑ,x0=x0,where xt∈Rn denotes the system state variable and ϑ∈Rl is the design parameter of system (12). System (12) is said to converge to the origin within a finite time if there exists a settling time function T:Rn⟶R+, such that, for every x0∈Rn, the solution of system (12) satisfies limt⟶Tx0Ψt,x0=0. For a globally finite-time convergent system (12), if for any initial state x0∈Rn, there exists a bounded convergence time Tmax irrelevant to initial conditions satisfying Tx0<Tmax, the origin has global fixed-time convergence.Definition 2.
(see [42–44]). The origin of the fixed-time stable system (12) is predefined-time stable if, for any convergence time Tc∈R+, there is ϑ∈Rl such that the settling-time function of (12) satisfies Tx0x0∈ℝn≤Tc.Definition 3.
(see [38]). Define a continuous function κ:R+⟶0,1. It belongs to class-▿, and there is κ∈▿1. κ is monotonically increasing such that κ0=0 and limr⟶∞κr=1.
Notion 1 (see [38]). For a given vector x=x1,x2,…,xnT, it can be obtained that sigax=xasignx and sigax=x1asignx1,x2asignx2,…,xnasignxnT, where signx denotes the sign function.Lemma 1.
(see [38]). For system (12), define an unbounded Lyapunov function Vx and let κ∈▿1 be differentiable in R\0. For any predefined convergence time Tc∈R+ and ϑt,x0∈Rl, the time-derivative of Vx satisfies(13)V˙x≤−11−pTcκVPκ′V,forx∈Rn\0,where 0<p<1, such that system (12) is predefined-time stable.Lemma 2.
[45]. Define a Lyapunov function Vx and let κV=Vξ/Vξ+χ, with χ>0, such that(14)V˙≤−1ξχ1−pTcVξp−ξ+1Vξ+χ2−p+Ω,for any solution ϑt,x0 of system (12), where Tc>0 denotes a predeﬁned-time parameter, χ>0 is a tuning parameter to adjust the convergence speed, Ω>0 is a small positive number, and 0<ξ<1 stands for an exponential coefficient. Then, system (12) origin has global predeﬁned-time stability, with predeﬁned time Tc.Proof.
Firstly, system (14) can be transformed into a differential equation:(15)V˙=−1ξχ1−pTcVξp−ξ+1Vξ+χ2−p.
By integrating equation (15), there is(16)Tx0=−∫Vx00dV1/ξχ1−pTcVξp−ξ+1Vξ+ϒ2−p=ξχ1−pTc∫0Vx0dVVξp−ξ+1Vμ+χ2−p=1−pTc∫0Vx0ξχVξ−1/Vξ+χ2Vξ/Vξ+χpdV=TcVξVξ+χ1−p|0Vx0.=Tc⋅Vξx0Vξx0+χ1−p
It is obvious thatlimVx0⟶∞Vξx0/Vξx0+χ=1 such that limVx0⟶∞Tx0=Tc. Therefore, the upper bound of the settling time is Tc.Theorem 1.
Define a radially unbounded Lyapunov functionVx and let κV=2/πarctanV such that(17)V˙≤−2πp−11+V21−pTcarctanVp+Ω,for any solution ϑt,x0 of system (12), where Tc>0, 0<p<1, and Ω>0 is a small positive number. Then, the origin of system (12) has predeﬁned-time stability, with predeﬁned time Tc.Proof.
Firstly, system (17) can be transferred as a differential equation:(18)V˙=−2πp−11+V21−pTcarctanVp.
By integrating with equation (18), there is(19)Tx0=−∫Vx00dV1/1−pTcπ1+V22/πarctanVp/2=21−pTcπ∫0Vx0dV1+V22/πarctanVp=1−pTc∫0Vx02/π1+V22/πarctanVpdV=Tc2πarctanV1−p|0Vx0=Tc⋅2πarctanVx01−p.
It is obvious thatlimVx0⟶∞2/πarctanVx0=1 such that limVx0⟶∞Tx0=Tc. Therefore, the bound of the settling time is Tc. Then, the origin of system (12) has predeﬁned-time stability, with predeﬁned time Tc.Lemma 3 (see [46]).
For anyyi∈R,i=1,2,…,n∑i=1nyip≤∑i=1nyip, where p∈R+ and p∈0,1.Lemma 4 (see [46]).
Ifη∈R+ and η>1, for any x,y∈R, there exists x+yη≤2η−1xη+yη.


## 3. Predefined-Time Cooperative Guidance

In this section, a novel predefined-time cooperative guidance law with impact angle constraints is designed, which consists of two parts: the cooperative guidance lawuri in the LOS direction and the cooperative guidance law uqi in the normal direction of LOS. And the strict proofs of the designed cooperative guidance law uri and uqi are given. The structure diagram of the predefined-time cooperative guidance system is shown in Figure 2.Figure 2
Structure diagram of the cooperative guidance system.


### 3.1. Guidance Law in the LOS Direction

From equation (5), the guidance system is formulated as(20)x˙1i=x2i,x˙2i=x1ix4i2−uri.The impact timetFi can be presented as(21)tFi=t+tgi.Assume each missile can simultaneously arrive.tgi will achieve a consensus when tFi is convergent. tFi can be rewritten as(22)t˙Fi=u˜riAccording to Definition1, under the protocol u˜ri, tFi can uniformly converge within the predefined time.Theorem 2.
Suppose that the undigraph of system (12) is connected. There is a consensus protocol as shown in equation (23) which can make system (12) achieve predefined-time stability:(23)u˜ri=2πδ−11+1/4∑i=1n∑j=1naijxj−xi221−δTc1⋅arctan1/4∑i=1n∑j=1naijxj−xi2δ∑i=1naijxj−xisign∑i=1naijxj−xi,where 0<δ<1.Proof.
Consider the following Lyapunov candidate:(24)Vx=14∑i=1n∑j=1naijxj−xi2.
Taking the derivative ofVx yields(25)V˙i=∑i=1n∂Vx∂xix˙i=−∑i=1naijxj−xi2πδ−11+1/4∑i=1n∑j=1naijxj−xi221−δTc1×arctan1/4∑i=1n∑j=1naijxj−xi2δ∑i=1naijxj−xisign∑i=1naijxj−xi=−∑i=1naijxj−xi2πδ−11+1/4∑i=1n∑j=1naijxj−xi221−δTc1×arctan1/4∑i=1n∑j=1naijxj−xi2δ∑i=1naijxj−xi≤−2πδ−11+1/4∑i=1n∑j=1naijxj−xi221−δTc1×arctan14∑i=1n∑j=1naijxj−xi2δ.
Therefore, equation (24) can be transformed into(26)V˙i≤−2πδ−11+1/4∑i=1n∑j=1naijxj−xi221−δTc1⋅arctan14∑i=1n∑j=1naijxj−xi2δ≤−2πδ−11+V21−δTc1⋅arctanVδ.
According to Theorem1, system (23) can achieve predefined-time stability by the proposed u˜ri within the predefined time Tc1, and this completes the proof of Theorem 2.Theorem 3.
Suppose that the undigraph is connected in the multiple-missile system. The guidance lawuri as shown in equation (27) can make uri achieve the predefined-time consensus:(27)uri=x1ix4i2−x2i2x1iu˜ri,where(28)u˜ri=2πδ−11+1/4∑i=1n∑j=1naijx1i/x2i−x1j/x2j221−δTc1⋅arctan1/4∑i=1n∑j=1naijaijx1i/x2i−x1j/x2j22δ∑i=1naijx1i/x2i−x1j/x2jsign∑i=1naijx1ix2i−x1jx2j.Proof.
Combine the derivative oftFi with equation (5), and we have(29)t˙Fi=x˙4i2x˙2i2−x1ix˙2i2uri=2πδ−11+1/4∑i=1n∑j=1naijtFj−tFi221−δTc1⋅arctan1/4∑i=1n∑j=1naijtFj−tFi2δ∑i=1naijtFj−tFisign∑i=1naijtFj−tFi.
According to Theorem2, tFi can reach a consensus if t>tFi. It demonstrates that each missile can exchange information in the guidance process and simultaneously intercept the target.Remark 1.
The proposed guidance lawuri ensures that tFi can reach a consensus within the predefined time Tc1 depending on the consensus protocol in equation (23). The convergence time can be adjusted based on the predefined time Tc1. In addition, by changing the magnitude of the design parameter δ, the convergence speed can be further improved.


### 3.2. Fixed-Time Disturbance Observer

The cooperative guidance model of equation (5) in the normal direction of LOS can be presented as follows:(30)x˙3i=x4i,x˙4i=−2x2ix1ix4i−uqix1i+Dqi,where Dqi=wqi/x1i denotes the disturbance caused by the target acceleration.However, the external disturbanceDqi is unmeasurable during the flight. In order to solve the problem, an FxTDO is used for the estimation of the external disturbance Dqi. Then, the estimated disturbance can compensate for the system. The disturbance observer (see [47]) is defined as(31)z˙1=z2+θε1ρx2−z1,η1,η2−2x2ix1ix4i−uqix1i,z˙2=θ2ε2ρx2−z1,η1,η2,where z1 and z2 are estimated values of x2 and the disturbance Dqi, respectively. η1, η2, φ, ε1, and ε2 are the design parameters, satisfying ε1,ε2>0, ε1≥2ε2, φ≥0, η1∈0.5,1, and η2∈1,2. ρ· as the correction term is defined by(32)ρx,η1,η2=sigη1x,x<1,sigη2x,x≥1.The stability proof of the above disturbance observer can be found in [47]. In conclusion, the observation errors of x2i−z1i and Dqi−z2i are fixed-time convergent to the origin, and the fixed time Tdi satisfies(33)Tdix≤4φ11−η1+1η2−1.Furthermore, the actual convergence time of the FxTDO is required to be smaller than the convergence time of the guidance system.


### 3.3. Guidance Law in the Normal Direction of LOS

A new predefined-time guidance law in the normal direction of LOS is proposed to ensure that each missile can achieve the desired LOS angle. Then, a new sliding mode surface is presented as follows:(34)si=x4i+γ0sigμp1−μ+1x3ix3iμ+ϒ2−p1,where γ0=1/μϒ1−p1Tc2, 1<ϒ<2, 0.5<μ<1, Tc2>0, and 0<p1<1. Therefore, we can obtain the following results.Theorem 4.
For the cooperative guidance model in (30), the sliding mode surface can be designed as (34). Then, x3i=0 and x4i=0 can be achieved within the predefined time Tc2.Proof.
Suppose the controller can guaranteesi=0 regardless of external disturbances. Then, we have(35)x4i+γ0sigμp1−μ+1x3ix3iμ+ϒ2−p1=0,x4i=−γ0sigμp1−μ+1x3ix3iμ+ϒ2−p1.
Consider a Lyapunov function candidate asV1i=x3i, and the time-derivative of V1i is(36)V˙1i=x˙3isignx3i=−γ0x3iμp1−μ+1x3iμ+ϒ2−p1=−1μϒ1−p1Tc2V1iμp1−μ+1V1iμ+ϒ2−p1.
Based on Lemma1, the proposed sliding mode surface is predefined-time stable. x3i⟶0 and x4i⟶0 are obtained in the predefined time Tc2. This completes the proof of Theorem 4.
With the sliding mode surface in equation (34), the predefined-time cooperative guidance law can be designed as(37)ui=1x1i11−p2Tc32πp2−1arctansip21+si2signsi−2x2ix1ix4i+z2i+Fi,where z2i is the output signal of the FxTDO. Fi represents the term related to the derivative of si. Then, we have(38)s˙=x˙4i+x4iγ0x3iμp1−μx3iμ+ϒ2−p1+γ0sigμp1−μ+1x3i2−p1x3iμ+ϒ1−p1μsigμ−1x3i,where(39)Fi=x4iγ0x3iμp1−μx3iμ+ϒ2−p1+sigμp1−μ+1x3i2−p1x3iμ+ϒ1−p1μx3iμ−1.Theorem 5.
If the PTCG law is designed as (37), then the sliding mode surface si=0 can be achieved within the predefined time.Proof.
Substituting equation (30) into s˙i yields(40)s˙i=ai+biuqi+Dqi+Fi,where ai=−2x2ix4i/x1i and bi=−1/x1i.
Bring equation (37) into equation (40), and we have(41)s˙i=ai+biuqi+Dqi+Fi=−2πp2−1arctansip21+si2signsi1−p2Tc3+ai+z2i+Fi,+ai+Dqi+Fi=−2πp2−1arctansip21+si2signsi1−p2Tc3−z2i+Dqi.
Considering the Lyapunov function,(42)V2i=si
Taking the derivative ofV2i yields(43)V˙2i=s˙isigns˙i=−2p2−1πp2−11+si21−p2Tc3arctansip2+Di−z2isignsi=−2p2−1πp2−11+V2i21−p2Tc3arctanV2ip2+Di−z2isignV2i.
For the disturbance observer in (31), the observation error Dqi−z2i will approach zero in a fixed time. Thus, there exists a convergence time Tdri, satisfying Tdri<Tc3, and we can get(44)Dqi−z2i≤N,t>Tdri,where N is a small constant.
Then, (43) can be simplified as(45)V˙2i≤−2p2−1πp2−11+si21−p2Tc3arctansip2+N≤−2p2−1πp2−11+V2i21−p2Tc3arctanV2ip2+Ω.
By utilizing Theorem1, the sliding mode surface s=0 can be reached within the predefined time Tc3. This finishes the proof of Theorem 5, which means that the design of the PTCG law for the multimissile system is completed. Consequently, the PTCG law can achieve predefined-time stability with the proposed ui in the predefined-time Tc=Tc2+Tc3.Remark 2.
By employing the FxTDO, the disturbance information can be obtained, which enhances the guidance accuracy of the given PTCG law. In fact, considering the convergence time of the observer can affect the convergence time of the whole system, the predefined convergence time of the guidance system should beTc=Tc2+Tc3+Tdri in the existence of uncertainties. And the system reaches the sliding mode surface s=0 within the predefined-time interval Tc3+Tdri. However, Tdri can be covered by Tc3 when the predefined time Tc3 is larger than the convergence time from the initial state of the system to the sliding surface according to Theorem 5. Furthermore, it is necessary to select an appropriate Tc2 or Tc3 as the upper bound. The adjustments of the guidance law parameters Tc2, Tc3, p1, and p2 are based on the simulation results.


## 3.1. Guidance Law in the LOS Direction

From equation (5), the guidance system is formulated as(20)x˙1i=x2i,x˙2i=x1ix4i2−uri.The impact timetFi can be presented as(21)tFi=t+tgi.Assume each missile can simultaneously arrive.tgi will achieve a consensus when tFi is convergent. tFi can be rewritten as(22)t˙Fi=u˜riAccording to Definition1, under the protocol u˜ri, tFi can uniformly converge within the predefined time.Theorem 2.
Suppose that the undigraph of system (12) is connected. There is a consensus protocol as shown in equation (23) which can make system (12) achieve predefined-time stability:(23)u˜ri=2πδ−11+1/4∑i=1n∑j=1naijxj−xi221−δTc1⋅arctan1/4∑i=1n∑j=1naijxj−xi2δ∑i=1naijxj−xisign∑i=1naijxj−xi,where 0<δ<1.Proof.
Consider the following Lyapunov candidate:(24)Vx=14∑i=1n∑j=1naijxj−xi2.
Taking the derivative ofVx yields(25)V˙i=∑i=1n∂Vx∂xix˙i=−∑i=1naijxj−xi2πδ−11+1/4∑i=1n∑j=1naijxj−xi221−δTc1×arctan1/4∑i=1n∑j=1naijxj−xi2δ∑i=1naijxj−xisign∑i=1naijxj−xi=−∑i=1naijxj−xi2πδ−11+1/4∑i=1n∑j=1naijxj−xi221−δTc1×arctan1/4∑i=1n∑j=1naijxj−xi2δ∑i=1naijxj−xi≤−2πδ−11+1/4∑i=1n∑j=1naijxj−xi221−δTc1×arctan14∑i=1n∑j=1naijxj−xi2δ.
Therefore, equation (24) can be transformed into(26)V˙i≤−2πδ−11+1/4∑i=1n∑j=1naijxj−xi221−δTc1⋅arctan14∑i=1n∑j=1naijxj−xi2δ≤−2πδ−11+V21−δTc1⋅arctanVδ.
According to Theorem1, system (23) can achieve predefined-time stability by the proposed u˜ri within the predefined time Tc1, and this completes the proof of Theorem 2.Theorem 3.
Suppose that the undigraph is connected in the multiple-missile system. The guidance lawuri as shown in equation (27) can make uri achieve the predefined-time consensus:(27)uri=x1ix4i2−x2i2x1iu˜ri,where(28)u˜ri=2πδ−11+1/4∑i=1n∑j=1naijx1i/x2i−x1j/x2j221−δTc1⋅arctan1/4∑i=1n∑j=1naijaijx1i/x2i−x1j/x2j22δ∑i=1naijx1i/x2i−x1j/x2jsign∑i=1naijx1ix2i−x1jx2j.Proof.
Combine the derivative oftFi with equation (5), and we have(29)t˙Fi=x˙4i2x˙2i2−x1ix˙2i2uri=2πδ−11+1/4∑i=1n∑j=1naijtFj−tFi221−δTc1⋅arctan1/4∑i=1n∑j=1naijtFj−tFi2δ∑i=1naijtFj−tFisign∑i=1naijtFj−tFi.
According to Theorem2, tFi can reach a consensus if t>tFi. It demonstrates that each missile can exchange information in the guidance process and simultaneously intercept the target.Remark 1.
The proposed guidance lawuri ensures that tFi can reach a consensus within the predefined time Tc1 depending on the consensus protocol in equation (23). The convergence time can be adjusted based on the predefined time Tc1. In addition, by changing the magnitude of the design parameter δ, the convergence speed can be further improved.


## 3.2. Fixed-Time Disturbance Observer

The cooperative guidance model of equation (5) in the normal direction of LOS can be presented as follows:(30)x˙3i=x4i,x˙4i=−2x2ix1ix4i−uqix1i+Dqi,where Dqi=wqi/x1i denotes the disturbance caused by the target acceleration.However, the external disturbanceDqi is unmeasurable during the flight. In order to solve the problem, an FxTDO is used for the estimation of the external disturbance Dqi. Then, the estimated disturbance can compensate for the system. The disturbance observer (see [47]) is defined as(31)z˙1=z2+θε1ρx2−z1,η1,η2−2x2ix1ix4i−uqix1i,z˙2=θ2ε2ρx2−z1,η1,η2,where z1 and z2 are estimated values of x2 and the disturbance Dqi, respectively. η1, η2, φ, ε1, and ε2 are the design parameters, satisfying ε1,ε2>0, ε1≥2ε2, φ≥0, η1∈0.5,1, and η2∈1,2. ρ· as the correction term is defined by(32)ρx,η1,η2=sigη1x,x<1,sigη2x,x≥1.The stability proof of the above disturbance observer can be found in [47]. In conclusion, the observation errors of x2i−z1i and Dqi−z2i are fixed-time convergent to the origin, and the fixed time Tdi satisfies(33)Tdix≤4φ11−η1+1η2−1.Furthermore, the actual convergence time of the FxTDO is required to be smaller than the convergence time of the guidance system.


## 3.3. Guidance Law in the Normal Direction of LOS

A new predefined-time guidance law in the normal direction of LOS is proposed to ensure that each missile can achieve the desired LOS angle. Then, a new sliding mode surface is presented as follows:(34)si=x4i+γ0sigμp1−μ+1x3ix3iμ+ϒ2−p1,where γ0=1/μϒ1−p1Tc2, 1<ϒ<2, 0.5<μ<1, Tc2>0, and 0<p1<1. Therefore, we can obtain the following results.Theorem 4.
For the cooperative guidance model in (30), the sliding mode surface can be designed as (34). Then, x3i=0 and x4i=0 can be achieved within the predefined time Tc2.Proof.
Suppose the controller can guaranteesi=0 regardless of external disturbances. Then, we have(35)x4i+γ0sigμp1−μ+1x3ix3iμ+ϒ2−p1=0,x4i=−γ0sigμp1−μ+1x3ix3iμ+ϒ2−p1.
Consider a Lyapunov function candidate asV1i=x3i, and the time-derivative of V1i is(36)V˙1i=x˙3isignx3i=−γ0x3iμp1−μ+1x3iμ+ϒ2−p1=−1μϒ1−p1Tc2V1iμp1−μ+1V1iμ+ϒ2−p1.
Based on Lemma1, the proposed sliding mode surface is predefined-time stable. x3i⟶0 and x4i⟶0 are obtained in the predefined time Tc2. This completes the proof of Theorem 4.
With the sliding mode surface in equation (34), the predefined-time cooperative guidance law can be designed as(37)ui=1x1i11−p2Tc32πp2−1arctansip21+si2signsi−2x2ix1ix4i+z2i+Fi,where z2i is the output signal of the FxTDO. Fi represents the term related to the derivative of si. Then, we have(38)s˙=x˙4i+x4iγ0x3iμp1−μx3iμ+ϒ2−p1+γ0sigμp1−μ+1x3i2−p1x3iμ+ϒ1−p1μsigμ−1x3i,where(39)Fi=x4iγ0x3iμp1−μx3iμ+ϒ2−p1+sigμp1−μ+1x3i2−p1x3iμ+ϒ1−p1μx3iμ−1.Theorem 5.
If the PTCG law is designed as (37), then the sliding mode surface si=0 can be achieved within the predefined time.Proof.
Substituting equation (30) into s˙i yields(40)s˙i=ai+biuqi+Dqi+Fi,where ai=−2x2ix4i/x1i and bi=−1/x1i.
Bring equation (37) into equation (40), and we have(41)s˙i=ai+biuqi+Dqi+Fi=−2πp2−1arctansip21+si2signsi1−p2Tc3+ai+z2i+Fi,+ai+Dqi+Fi=−2πp2−1arctansip21+si2signsi1−p2Tc3−z2i+Dqi.
Considering the Lyapunov function,(42)V2i=si
Taking the derivative ofV2i yields(43)V˙2i=s˙isigns˙i=−2p2−1πp2−11+si21−p2Tc3arctansip2+Di−z2isignsi=−2p2−1πp2−11+V2i21−p2Tc3arctanV2ip2+Di−z2isignV2i.
For the disturbance observer in (31), the observation error Dqi−z2i will approach zero in a fixed time. Thus, there exists a convergence time Tdri, satisfying Tdri<Tc3, and we can get(44)Dqi−z2i≤N,t>Tdri,where N is a small constant.
Then, (43) can be simplified as(45)V˙2i≤−2p2−1πp2−11+si21−p2Tc3arctansip2+N≤−2p2−1πp2−11+V2i21−p2Tc3arctanV2ip2+Ω.
By utilizing Theorem1, the sliding mode surface s=0 can be reached within the predefined time Tc3. This finishes the proof of Theorem 5, which means that the design of the PTCG law for the multimissile system is completed. Consequently, the PTCG law can achieve predefined-time stability with the proposed ui in the predefined-time Tc=Tc2+Tc3.Remark 2.
By employing the FxTDO, the disturbance information can be obtained, which enhances the guidance accuracy of the given PTCG law. In fact, considering the convergence time of the observer can affect the convergence time of the whole system, the predefined convergence time of the guidance system should beTc=Tc2+Tc3+Tdri in the existence of uncertainties. And the system reaches the sliding mode surface s=0 within the predefined-time interval Tc3+Tdri. However, Tdri can be covered by Tc3 when the predefined time Tc3 is larger than the convergence time from the initial state of the system to the sliding surface according to Theorem 5. Furthermore, it is necessary to select an appropriate Tc2 or Tc3 as the upper bound. The adjustments of the guidance law parameters Tc2, Tc3, p1, and p2 are based on the simulation results.


## 4. Simulations and Results

To verify the effectiveness of the given PTCG, the case where three missiles intercept a moving target is simulated. The initial values of the target are set asVT=300m/s and aT=45costm/s2. The communication topology of the missiles is shown in Figure 3, and the weighted adjacency matrix of the communication topology is A=010;101;010. The lateral acceleration of three interceptors satisfies aimax=200m/s2. The initial values of this scenario are given in Table 1.Figure 3
Communication topology among the missiles.Table 1
Initial values of the missiles and target.
MissileInitial position (m, m)Heading angle (°)Desired angle (°)Initial velocity (m/s)M1(5940, −1040)11−6610M2(4900, −850)−100610M3(4105, 995)−105600Target(0, 0)60—300The parameters of the proposed PTCG law and FxTDO are given in Table2.Table 2
Parameters of the PTCG.
TcTc1δp1μγTc2p2Tc3η1η2ε1ε2θ9.51.50.950.80.81.64.50.450.71.510203Simulation results of the comparative experiment by the proposed PTCG law and by the finite-time cooperative guidance law in [27] in the same circumstance are presented in Figures 4–11 and Table 3.Figure 4
Curves of relative distance.Figure 5
Curves of LOS angle.Figure 6
Curves of LOS angular velocity.Figure
Curves of sliding mode surface.Figure 8
Acceleration commanduqi.Figure 9
Time-to-go for three missiles.Figure 10
Acceleration commanduri.Figure 11
Disturbance estimation results.Table 3
Final parameters.
MissileM1M2M3PTCGMiss distance (m)0.050.090.16Impact LOS angle (°)5.0010.001−6.001Interception time (s)18.7118.7118.71FTCGMiss distance (m)0.340.470.45Impact LOS angle (°)5.190.17−6.13Interception time (s)21.2621.2621.26From Figure4 and Table 3, it is demonstrated that both PTCG and FTCG can intercept the maneuvering target simultaneously. Furthermore, the proposed PTCG has higher precision and shorter interception time than the FTCG in the mission.As shown in Figure5, the LOS angles qi can converge to the desired values by these two cooperative guidance laws. And all, the LOS rates can approach zero as shown in Figure 6. More importantly, Figures 5 and 6 also show that the convergence times of qi and qdi are both less than 5 s, and the convergence curves are smoother under the PTCG. Meanwhile, it is observed that the given PTCG can ensure that the guidance system achieves predefined-time stability within 9.5 s. In terms of the convergence rate, PTCG converges faster than FTCG.As illustrated in Figure7, the sliding mode surface of the PTCG achieves predefined-time stability within 4.5 s. Hence, the predefined-time stability of the given PTCG law is verified by the above results, and the proposed PTCG performs better than the FTCG. Although the sliding dynamics is broken in the terminal flight phase, that is, the absolute value of the sliding mode surface does not equal zero, it owns to the finite sampling frequency of the mathematical simulation. When the relative distance between missile and target turns to a small amount, the LOS angle velocity becomes large, which is indicated from equation (30), and the absolute value of the sliding mode variable increases.In Figure8, a relatively large acceleration command uqi for the three missiles was given in the initial phase of flight, since a larger acceleration command can drive q˙ to zero at a faster rate so that q can converge to the desired value more quickly. As shown in Figure 9, by using the PTCG, it takes around 0.4 s for tgi to reach a consensus, which is faster than using the FTCG. Furthermore, Figure 9 demonstrates that the PTCG can achieve predefined-time stability within 1.5 s.The acceleration commanduri is shown in Figure 10. uri is utilized to regulate tgi of the multiple missiles so that tgi realizes uniform convergence within a predefined time. uri is quite large at the beginning, which reflects the fact that the less tgi it takes for the system to reach a consensus, the more energy it needs. In Figure 11, it is observed that the external disturbance is estimated accurately by the FxTDO, and all convergence time Tdri is less than 0.2 s. In conclusion, the proposed PTCG law enjoys predefined-time convergence compared with the FTCG law. Hence, the guidance system has a faster convergence rate and higher precision by the proposed PTCG.


## 5. Conclusion

In this paper, multiple missiles intercepting a maneuvering target have been solved by the proposed PTCG law. Under a novel predefined-time consensus protocol, a PTCG law along the LOS direction is designed. The designed PTCG law enables the time-to-go of the multiple missiles to reach a consensus in the predefined time. Considering the disturbance caused by the target maneuver, a PTCG law with an FxTDO in the normal direction of the LOS direction is presented. Thus, the guidance system can guarantee predefined-time stability. Meanwhile, strict proof of predefined-time stability for the proposed PTCG law is presented. Finally, the designed PTCG law was validated by numerical simulations, which indicates the strong robustness and adaptability of the proposed PTCG. In the future work, we will further improve the proposed algorithm by considering switching communication topologies and interception of hypersonic vehicles in the design of the cooperative guidance law.



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*Source: 9940240-2021-07-06.xml*