JCSEJournal of Control Science and Engineering1687-52571687-5249Hindawi10.1155/2020/10583471058347Research ArticleAdaptive Super-Twisting Algorithm-Based Nonsingular Terminal Sliding Mode Guidance LawYangFang1https://orcid.org/0000-0002-1912-5001ZhangKuanqiao2YuLei2WangYu1Xi’an Aeronautical UniversityXi’an 710077Chinaxaau.edu.cn2Luoyang Electronic Equipment Test Center of ChinaLuoyang 471003China202047202020202212020285202011620204720202020Copyright © 2020 Fang Yang et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A nonsingular fast terminal sliding mode guidance law with an impact angle constraint is proposed to solve the problem of missile guidance accuracy and impact angle constraint for maneuvering targets. Aiming at the singularity problem of the terminal sliding mode, a fast terminal sliding mode surface with finite-time convergence and impact angle constraint is designed based on fixed-time convergence and piecewise sliding mode theory. To weaken chattering and suppress interference, a second-order sliding mode supertwisting algorithm is improved. By designing the parameter adaptive law, an adaptive smooth supertwisting algorithm is designed. This algorithm can effectively weaken chattering without knowing the upper bound information of interference, and it converges faster. Based on the proposed adaptive supertwisting algorithm and the sliding mode surface, a guidance law with the impact angle constraint is designed. The global finite-time convergence of the guidance law is proved by constructing the Lyapunov function. The simulation results verify the effectiveness of the proposed guidance law, and compared with the existing terminal sliding mode guidance laws, the proposed guidance law has higher guidance precision and angle constraint accuracy.

1. Introduction

In modern warfare, many missiles (such as some antiship missiles, antitank missiles, and air defense missiles) need to hit the target with certain impact angles to increase the damage effectiveness of the warheads. Therefore, the impact angle constraint is a problem that needs to be considered in the design of the guidance law .

Sliding mode control is widely used in the design of the guidance law because of its invariability to interference in the sliding mode. In , the line-of-sight (LOS) angular velocity and impact angle constraint have been used as the sliding surface, and the sliding mode control is applied to design the guidance law with the impact angle constraint. In , the adaptive exponential approach law has been used to design the sliding mode guidance law, which increases the adaptability and dynamic performance of the guidance law. However, none of these methods has finite-time convergence. For the finite-time control problem, a finite-time convergence guidance law with the impact angle constraint is designed based on the terminal sliding mode control in . In [7, 8], linear terms are added to the terminal sliding mode surface to further speed up the convergence of system states. But the negative exponential term of the state quantity in the terminal sliding mode control law will cause the singular problem. For the singular problem, a nonsingular terminal sliding surface is improved to avoid the singularity problem in , and the corresponding guidance law is designed. However, the proposed guidance laws cannot guarantee strictly finite-time convergence of the sliding mode surface, and there are nonconvergence factors, so the convergence rate will be reduced. In , the problem of nonstrict convergence of the sliding surface is studied, and a nonsingular terminal sliding surface with strictly finite-time convergence is proposed. However, the sliding surface function is not smooth, the system can only converge to a bounded region, and the specific range cannot be given.

For the disturbance problems such as target maneuvering and system disturbance, there are currently three methods for processing most documents: (1) designing the disturbance observer to estimate the disturbance in real time and online [13, 14]; (2) designing the adaptive law to estimate the upper bound of disturbance [15, 16]; and (3) using the robustness of the sliding mode control to resist interference. These methods need to introduce symbolic function terms which will make the control quantity discontinuous and easy to cause the chattering phenomenon. Most literature studies smooth the symbol terms to reduce chattering. However, at the same time, they also change the inherent structure of the sliding mode control and weaken the robustness of the sliding mode control system. For the chattering problem, a second-order sliding mode supertwisting algorithm is proposed in . It has the advantages of simple form, avoiding chattering, and strong robustness. However, the control law of the supertwisting algorithm is not smooth, the parameter selection needs to know the upper boundary information of system disturbance, and the convergence speed is slow when the system states are far from the equilibrium point.

In order to solve the above problems, this paper improves an adaptive smooth supertwisting algorithm, which solves the problems of slow convergence speed and the unsmooth control law of the traditional supertwisting algorithm and greatly weakens the chattering problem of the sliding mode control. At the same time, the parameter adaptive law is designed against the disturbance without knowing the upper bound information of the disturbance. Based on the idea of fixed-time convergence and piecewise sliding surface, a nonsingular fast terminal sliding surface with the impact angle constraint is designed. A nonsingular fast terminal sliding mode guidance law with the impact angle constraint is proposed based on the adaptive supertwisting algorithm. The global finite-time convergence is proved by constructing the Lyapunov function. Finally, the effectiveness and superiority of the guidance law are verified by simulation experiments.

2. Preparatory Knowledge2.1. Relative Dynamics between the Missile and the Target

In the inertial coordinate system, the relative motion relationship between the missile and the target is established as shown in Figure 1. M and T represent the missile and the target, respectively. r is the relative distance between the missile and the target, and q is the LOS angle. vm and vt are velocities of the missile and the target, respectively, and θm and θt are track angles of the missile and the target, respectively.

Relationship of missile-to-target motion.

According to the relative motion relationship of the missile and the target, the relative motion equations of the missile and the target can be obtained as follows:(1)r˙=vmcosqθm+vtcosqθt,rq˙=vmsinqθmvtsinqθt,am=vmθ˙m,at=vtθ˙t.

Differentiating q˙ with respect to time gives(2)q¨=2r˙q˙ramcosqθmr+atcosqθtr.

The impact angle is the angle between the missile and the target velocity vector at the time of guidance terminal, and the impact angle constraint problem can be transformed into the terminal LOS angle constraint problem . Therefore, the state equation of the guidance system with the impact angle constraint can be obtained based on (2) as follows:(3)x˙1=x2,x˙2=f1x2+f2u+d,with(4)x1=qqd,x2=q˙,u=am,f1=2r˙r,f2=1r.d=amamcosqθm+atcosqθtr,where qd is the desired terminal LOS angle; d can be regarded as the total disturbance of the system.

2.2. Related Lemma

For the convenience of analysis and proof, the following lemmas are introduced.

Lemma 1 (see [<xref ref-type="bibr" rid="B18">18</xref>]).

Assume that there is a smooth function Vx defined on the neighborhood U^U0Rn of the origin, and a1 > 0 and 0 < b1 < 1, V˙x+a1Vb1x0; then, the origin of the system is finite-time stable, and the convergence time satisfies(5)T1V1b1x0a11b1.

Lemma 2 (see [<xref ref-type="bibr" rid="B19">19</xref>]).

Assume that Lyapunov function Vx satisfies V˙xa1Vb1xa2Vx, and a1 > 0, a2 > 0, and 0 < b1 < 1; then, the system can converge to the origin in finite time, and the convergence time satisfies(6)T21a21b1ln1+a2V1b1x0a1.

Lemma 3 (see [<xref ref-type="bibr" rid="B20">20</xref>, <xref ref-type="bibr" rid="B21">21</xref>]).

For the nonlinear system y˙=a1yb1sgnya2yb2sgny, if a1 > 0, a2 > 0, 0 < b1 < 1, and b2 > 1, the system is stable in finite time, and the convergence time satisfies(7)T3<1a11b1+1a2b21.

In addition, if the system has a small disturbance, that is, y˙=a1yb1sgnya2yb2sgny+ς and ς is a small positive number, the system can converge to the neighborhood Ω=y2ϑa1ϑb1+a2ϑb2=ς of the origin in finite time, and the convergence time satisfies(8)T4<1a12b111b1+1a2b21.

For the following first-order system,(9)y˙=u+ξ,where y is the system state, u is the input, and ξ is the disturbance, the supertwisting algorithm can be expressed as follows:(10)u=m1y1/2sgny+u1,u˙1=m2sgny.

The supertwisting algorithm can greatly reduce chattering and has strong robustness and high precision control performance . However, the supertwisting algorithm has the following disadvantages: (1) the control law is a continuous function, but not a smooth function, which will affect the control performance; (2) the selection of control parameters needs to know the upper bound information of the system disturbances; and (3) when the system states are far from the equilibrium point, the convergence speed is slow. In view of the above shortcomings, this paper speeds up the convergence of the algorithm by adding linear terms to the algorithm. And the adaptive law does not need the information of the interference. The improved adaptive supertwisting algorithm can be expressed as follows:(11)u=m1φ1y+u1,u˙1=m2φ2y,φ1y=1βyβsgny+y,φ2y=y+y|2β1sgnyβ+1β+1yβsgny,where 1/2β<1.

The parameter adaptive law is designed as follows:(12)m˙1=a+cφ1yφ2ysgnyε,m2=bm1,m10>0,where a>0, b>0, c>0, and ε>0.

Substituting (11) into (9),(13)y˙=m1φ1y+u1+ξ,u˙1=m2φ2y.

Remark 1.

It can be seen from (13) that when the system state is far away from the equilibrium point, the linear term y in (11) will accelerate the convergence rate of the system. When the system is close to the equilibrium point, the nonlinear term yβsgny plays an important role in accelerating the convergence rate of the system. Therefore, compared with the traditional supertwisting algorithm, the improved adaptive supertwisting algorithm (11) has a faster convergence speed.

Remark 2.

Due to the measurement noise of the system, the state of the system cannot reach the equilibrium point completely. In order to avoid the parameter increasing to infinity, the term sgnyε is added to the adaptive law to avoid the problem of overestimation .

Remark 3.

It is obvious that φ1sφ2s0. So, it can be seen from (12) that when y>ε, m1 and m2 will gradually increase, making the system state to converge. When the system state converges to y<ε, m1 and m2 will decrease gradually. If m1 and m2 decrease to the point where the interference cannot be eliminated, the system state will deviate from y<ε. At this point, m1 and m2 will gradually increase under the effect of the adaptive law, making the system state converge to y<ε. Repeat the previous process; m1 and m2 will gradually decrease. Therefore, m1 and m2 are globally bounded.

For the total disturbance ξ of the system, the following assumption can be made:

Assumption 1.

ξ is bounded, ξ=ξ1+ξ2, ξ1 is nondifferentiable, and ξ2 is differentiable; they satisfy(14)ξ1Kφ1y,ξ2Lφ2y.

Theorem 1.

Under Assumption 1, the existence of m¯1 makes the system state converge in finite time when m1m¯1 and m2=bm1.

Proof.

Define a new state vector as(15)z=z1z2=φ1ym20tφ2ydt+d2.

From (14), it can be seen that the existence of ρ1t and ρ2t makes the following equation valid:(16)ξ1=ρ1z1,ρ1K,ξ˙2=yβ1+1ρ2z1,ρ2L.

Differentiating (15) with respect to time gives(17)z˙=z˙1z˙2=yβ1+1m1z1+ρ1z1+z2ρ2z1m2z1=yβ1+1Az,with(18)A=m1+ρ11m2+ρ20.

Define the following Lyapunov function:(19)V1=zTPz,with(20)P=a2+b2bb1.

It is easy to prove that P is a positive definite matrix; then, V1 is unbounded radially, i.e.,(21)λminPz2V1λmaxPz2.

Differentiating V1 with respect to time gives(22)V˙1=yβ1+1zTATP+PAz=yβ1+1zTQz,with(23)Q=Q1Q2Q22b,Q1=am1+2bbm1m2ρ1a+2b2+2bρ2,Q2=m2bm1a2+b2+bρ1ρ2.

If we define(24)m¯1=ρ1a+2b2+2bρ2a+bρ1ρ2a/2b222ab,m1m¯1,m2=bm1,it can be proved that Q is a positive definite matrix, and λminQb.

According to (21), we can get(25)yzV1λminP1/2,(26)V˙1V1λminP1β/2+1λminQz2bλmin1β/2PV11+β/2λmaxPbλmaxPV1.

According to Lemma 2, z is finite-time convergent. The proof is complete.

Theorem 2.

Under the control law (11) and the control parameters which satisfy (12), the system state can converge to yε in finite time.

Proof.

According to (12), m2=bm1. If m1m¯1, then from Theorem 1, the system is finite-time convergent. If m1<m¯1, define the Lyapunov function as(27)V2=zTPzVz+m˜122Vm1,where m˜1=m1m¯1.

Differentiating V2 with respect to time gives(28)V˙2=yβ1+1zTATP+PAz+m˙1m˜1=yβ1+1zTA¯TP+PA¯zV˙z+yβ1+1zTA˜TP+PA˜z+m˙1m˜1V˙m1,with(29)A¯=m¯1+ρ11bm¯1+ρ20,A˜=AA¯=m˜10bm˜10.

According to (26), we can get(30)V˙zbλmin1β/2PVz1+β/2λmaxPbλmaxPVz.

When y>ε, combining with (12) gives(31)V˙m1=yβ1+1zTA˜TP+PA˜z+m˙1m˜1=aφ1yφ2y+a+cφ1yφ2ym˜1=cφ1yφ2ym˜1=2cφ1yφ2yVk1/2,V˙2bλmin1β/2PVz1+β/2λmaxPbλmaxPVz2cφ1yφ2yVk1/2=μV2γ,with(32)μ=μbλmin1β/2PVz1+β/2λmaxP,bλmaxP,2cφ1yφ2y>0.(33)γ=γ1+β2,1,12.

It is known that 1/2<γ<1 from 1/2β<1, so(34)V˙2μV2γ0.

According to Lemma 1, V2 can converge in finite time, y can converge to yε, and the convergence time satisfies(35)t1V21γ0μ1γ.

When yε, if m1 and m2 decrease to the point where the interference cannot be eliminated, the system state will deviate from yε. In this case, m1 and m2 will increase again under the effect of the adaptive law, making the system state converge to yε.

4. Guidance Law Design

The terminal sliding mode control adopts the nonlinear function as the sliding mode surface, which can make the system states converge in finite time, but the method has singular problems. In order to avoid singular problems, based on the piecewise sliding surface  and Lemma 3, a nonsingular fast terminal sliding surface is designed as(36)s=x2+k1x1α1sgnx1+k2ψx1,with(37)ψx1=x1α2,x1δ,gx1,x1<δ,where α1>1, 0<α2=p1/p2<1, k1>0, k2>0, δ>0, and p1 and p2 are positive odd numbers.

Differentiating s with respect to time gives(38)s˙=x˙2+k1α1x1α11x2+k2ψx1x2,with(39)ψx1=α2x1α21,x1δ,gx1,x1<δ,where gx1 is a function of x1 and satisfies the following conditions:

gx1 is a smooth function in x1δ,δ with the same sign as x1

gδ=ψδ=gδ

gδ=gδ=ψδ, and gx1>0 in x1δ,δ

Remark 4.

Condition (1) can ensure that gx1 is a continuous bounded function and eliminates singular problems, and when the system reaches the sliding surface s=0, x1 and x2 are always with different signs, ensuring that the system state is convergent. Condition (2) ensures that the sliding surface s is a continuous function. Condition (3) guarantees that gx1 is bounded in x1δ,δ, and ψx1 is a continuous function, so s is a smooth function.

According to the above conditions, this paper selects function gx1 as follows:(40)gx1=λ1x1+λ2x13,where λ1=3α2/2εα21 and λ2=α21/2εα23.

Substituting (3) into (38) yields(41)s˙=f1x2+f2u+d+k1α1x1α11x2+k2ψx1x2.

The equivalent guidance law is designed as(42)ueq=f21f1+k1α1x1α11+k2ψx2.

Substituting (42) into (41) gives(43)s˙=d.

In order to counteract the disturbance, suppress chattering, and accelerate the convergence speed of the sliding surface, based on the adaptive smooth, fast supertwisting algorithm proposed in the second section, an auxiliary guidance law is designed as(44)uaux=f21k3φ1s+k40tφ2sdt.

The parameter adaptive law is designed as(45)k˙3=a+cφ1sφ2ssgnsε,k4=bk3,k30>0.

Combining with (42) and (44), we can design a nonsingular terminal sliding mode guidance law with the impact angle constraint based on the adaptive supertwisting algorithm as(46)u=ueq+uaux=f21f1+k1α1x1α11+k2ψx2+k3φ1s+k40tφ2sdt.

For the convenience of description, the design guidance law (46) is abbreviated as ASNTSMG.

5. Simulation Analysis

In order to test the performance of the designed guidance law, ASNTSMG, this section conducts simulation analysis based on ballistic simulation in different scenarios. The initial position of the missile is (0 m, 0 m), and the initial position of the target is (1000 m, 5000 m). The missile’s velocity is vm = 500 m/s, and the target’s velocity is vt = 250 m/s. The acceleration of gravity is g = 9.8 m/s2, and the maximum available overload of the missile is 20 g. The parameters of ASNTSMG are set as follows: k1=k2=2, a1=3, α2=5/7, ε=0.01, δ=0.001, a=0.5, and b=c=1.

In order to verify the superiority of the designed guidance law, this section also carries out the nonsingular fast terminal sliding mode guidance law (NFTSMG) proposed in  and the second-order nonsingular terminal sliding mode guidance law (SONTSMG) proposed in  to perform a comparative simulation. The expression of NFTSMG is(47)u=rcosqθm1k2a2x22a21+k1a1x1a112r˙rx2+αs+βsγsgnsr.

The parameters are set as follows: α=600, β=500, α1=7/5, α2=5/7, k1=k2=2, and γ=0.5.

The expression of SONTSMG is(48)u=2r˙rx2+rβαx22α+z1+k1s11/γsgns+k20tx2α1rs12/γsgnsdt.

The parameters are set as follows: k1=600, k2=100, α=7/5, β=0.5, and γ=2.1.

The average overload Nme (unit: g) is introduced to evaluate the energy consumption in the process of guidance, which is defined as follows:(49)Nme=1Ki=1Kami,where K is the total number of simulation steps.

Case 1.

Attack moving target with different impact angle constraints: set qd as 20°, 30°, 40°, and 50°, respectively, and θm0 = 45°. The target makes sinusoidal maneuver, and its acceleration is at = 30sin (πt/5) m/s2, and θt0 = 150°. The simulation results are shown in Figure 2.

It can be seen from Figures 2(a) and 2(b) that ASNTSMG can effectively intercept the target with different impact angle constraints. The miss distances are 0.374 m, 0.428 m, 0.408 m, and 0.479 m, respectively. This method can hit the target accurately. It can be seen from Figures 2(c)2(e) that the sliding surface and the LOS angular rate can converge to zero in finite time, and LOS angle can effectively converge to the expected impact angle. With the increase of qd, the convergence time increases. This is because the larger qd is, the larger the initial LOS angle deviation will be, and the convergence time is related to the initial value, which leads to the corresponding growth of the convergence time.

Figure 2(f) shows the overload curve of the missile, which is saturated in the early stage, and the larger θm0, the longer the saturation time, which is mainly due to the larger overload needed in the earlier stage, which makes the missile meet the requirements of angle constraint and guidance accuracy. When q and q˙ approach the expected values, the overload gradually approaches zero, which ensures that the missile has sufficient overload margin to deal with other unknown disturbances in the later stage of guidance.

Simulation results of Case 1. (a) Missile and target motion trajectory. (b) Relative distance between the missile and the target. (c) Sliding mode. (d) LOS angle. (e) LOS angular rate. (f) Missile overload.

Case 2.

Comparative simulation of ASNTSMG, NFTSMG, and SONTSMG: the relevant initial parameters are set to θm0 = 45°, qd=45°, and θt0 = 180°. The movement of the target is set as follows:

Cosine motion: at = 30cos (πt/5) m/s2

Square wave motion: at = 30sgn (sin (πt/5)) m/s2

The simulation results are shown in Figures 36 and Table 1.

Figure 3 shows the trajectories of the missile and the target. It can be seen that the missile can track and intercept the target under the three guidance laws. Compared with NFTSMG and SONTSMG, the trajectory of ASNTSMG is relatively smooth, indicating that its attack time is relatively short, which can be verified by Table 1. Figure 4 shows the LOS angle curve. All three guidance laws can make the LOS angle gradually converge to the expected angle. ASNTSMG can make the LOS angle converge to the expected angle more quickly. NFTSMG adopts the robustness of the sliding mode control to cancel the disturbance of target maneuver, so it can only make the system states converge to the neighborhood of origin in finite time. NFTSMG and SONTSMG adopt the traditional nonsingular terminal sliding surface, which has the nonstrict finite-time convergence problem, and the nonconvergence factor slows down the convergence speed. Figure 5 shows the LOS angular rate curve. Under the three guidance laws, the LOS angular rates converge to zero in finite time. ASNTSMG has smaller convergence error and faster convergence speed. Figure 6 shows the overload curve of the missile. Due to the finite-time convergence of LOS angle and angular rate, the missile needs a large overload in the early stage of guidance. Therefore, the overloads of three guidance laws are all saturated in the early stage. When the LOS angle and the angular rate converge, the missile overload gradually converges to zero in the later stage. And the convergence speed of ASNTSMG is faster.

Table 1 shows the simulation results of attack time, miss distance, LOS angle error, and average overload under the three guidance laws. It can be seen that compared with NFTSMG and SONTSMG, ASNTSMG has smaller attack time, miss distance, terminal LOS angle error, and average overload, so ASNTSMG has better guidance performance.

According to the analysis of the simulation results of two cases, ASNTSMG can hit the target precisely with the expected impact angle under different expected LOS angles and target maneuvering conditions. Compared with the existing guidance laws NFTSMG and SONTSMG, ASNTSMG can effectively attack the target with less impact angle error, miss distance, and energy consumption, which verifies the effectiveness and superiority of ASNTSMG.

Missile and target motion trajectory. (a) Target cosine motion. (b) Target square wave motion.

LOS angle. (a) Target cosine motion. (b) Target square wave motion.

LOS angular rate. (a) Target cosine motion. (b) Target square wave motion.

Missile overload. (a) Target cosine. (b) Target square wave motion.

Simulation results of different guidance laws.

Target movementGuidance lawAttack time (s)Miss distance (m)Angle error (deg)Nme (g)
Cosine maneuverNFTSMG16.240.950.593.88
SONTSMG16.120.760.213.58
ASNTSMG16.050.390.023.43

Square wave maneuverNFTSMG15.541.030.883.99
SONTSMG15.470.860.583.48
ASNTSMG15.410.430.033.36
6. Conclusion

In this paper, a nonsingular fast terminal sliding mode guidance law is proposed to solve the problem of guidance accuracy and impact angle constraint. Through theoretical analysis and simulation verification, the following conclusions can be obtained:

The proposed adaptive smooth supertwisting algorithm can effectively counteract the disturbance of the system and accelerate the convergence speed of the system without knowing the upper bound of the disturbance.

The designed nonsingular terminal sliding mode surface can realize the fast finite-time convergence of the system states and ensure the impact angle constraint and guidance accuracy requirements.

This guidance law can attack the target precisely under the conditions of different expected LOS angles and target maneuvers. Compared with the existing nonsingular fast terminal sliding mode guidance law and second-order nonsingular terminal sliding mode guidance law, this law has higher guidance accuracy and angle constraint accuracy and consumes less energy.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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