Non-singular terminal dynamic surface control based integrated guidance and control design and simulation

ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Non-singular terminal dynamic surface control based integrated guidance and control design and simulation Zhang Cong a,b,c, Wu Yun-jie a,b,c a

State Key Laboratory of Virtual Reality Technology and Systems, Beihang University, Beijing 100191, China School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China c Science and Technology on Aircraft Control Laboratory, Beihang University, Beijing 100191, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 1 February 2016 Received in revised form 25 February 2016 Accepted 14 March 2016

In this paper, a novel cascade type design model is transformed from the simulation model, which has a broader scope of application, for integrated guidance and control (IGC). A novel non-singular terminal dynamic surface control based IGC method is proposed. It can guarantee the missile with multiple disturbances fast hits the target with high accuracy, while considering the terminal impact angular constraint commendably. And the stability of the closed-loop system is strictly proved. The essence of integrated guidance and control design philosophy is reached that establishing a direct relation between guidance and attitude equations by “intermediate states” and then designing an IGC law for the obtained integrated cascade design model. Finally, a series of simulations and comparisons with a 6-DOF nonlinear missile that includes all aerodynamic effects are demonstrated to illustrate the effectiveness and advantage of the proposed IGC method. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Integrated guidance and control (IGC) Non-singular terminal sliding mode control (NTSMC) Dynamic surface control (DSC) Simulations

1. Introduction By accounting for the synergism between the guidance and control dynamics, IGC has the potential to improve performance during the terminal guidance phase and yield acceptable miss distances [1]. And the stability of IGC can be strictly proved by Lyapunov theory. Due to the wide array of potential benefits that an IGC approach can involve, there has been much interest in this area over the past few decades. The ideology of integrated guidance and control was put forward by Williams D E [2,3] in the early stage. Then it was mainly used to solve the homing guidance problems in the development of the past few decades. Various algorithms were used to drive it, such as linearization-LQR [4], adaptive control, robust control, sliding mode control (SMC) [5,6], adaptive SMC, feedback linearization, active disturbance rejection control (ADRC) [7], back-stepping approach [8,9], adaptive dynamic surface control [10,11] and so on. After that IGC was also used in other areas such as reactive obstacle avoidance of UAVs [12], formation flight [13,14], automatic landing of UAVs [15], cooperative attack of multiple missiles [16], reentry process [17,18] and so on. The existing researches involve dealing with IGC design either in the presence of longitudinal missile model only [19–21], or with limited design model [10,11], or without considering impact angle constraints [1,10], or without considering the ‘explosion of complexity’ of the back-stepping [7–9].

Refs. [10,11] are more relevant to this paper, an integrated guidance and autopilot design method was proposed for homing missiles based on the adaptive block dynamic surface control approach. But its assumption that the missile flies heading to the target at initial time cannot be satisfied in all case. So its design and model is very limited. The main contributions in this paper are summarized as follows: (1) A novel IGC design model is established, which has a wider scope of application than the model in [10,11]. It does not need the assumption that the missile flies heading to the target at the initial stage. And the essence of IGC is summarized. (2) A novel non-singular terminal dynamic surface control (NT_DSC) IGC law is designed combining non-singular terminal sliding mode control (NTSMC) [22,23] and dynamic surface control (DSC) [24]. It can guarantee the missile hits the target with higher accuracy while better satisfying the terminal impact angular constraints. And the practical stability of the overall system is strictly proved based on the Lyapunov stability theorem. This paper is organized as follows. A novel IGC design model is established and the essence of IGC is summarized in Section 2. NT_DSC IGC law is proposed in Section 3. In Section 4, simulations and comparisons are shown. The paper ends with a few conclusions in Section 5.

http://dx.doi.org/10.1016/j.isatra.2016.03.013 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Cong Z, Yun-jie W. Non-singular terminal dynamic surface control based integrated guidance and control design and simulation. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.03.013i

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2

2. Modeling In this section, the simulation model and a novel IGC design model are established. And the design objective of IGC law is elaborated. 2.1. Simulation model A-XYZ in Fig. 1 shows the inertial coordinate system. O-xyz is fixed to the missile centroid and parallel to A-XYZ. Firstly, rotate Oxyz counterclockwise by q2 along O-y axis, and then rotate it counterclockwise by q1 along O-z axis. One gets line-of-sight (LOS) coordinate system O-x4y4z4. In Fig. 1 q1 is elevation angle; q2 is azimuth angle; R is missiletarget relative distance. The guidance equations of the pursuit situation in the threedimensional space as shown in Fig. 1 can be described by the following nonlinear Eqs. (1) [10,11] 3 " # 2 2R_  R q_ 1  q_ 2 2 sin q1 cos q1 q€ 1 5 ¼4 _ q€ 2  2RRq_ 2 þ 2q_ 1 q_ 2 tan q1 2 3" # " # aty4  1R 0 ay4 R 5 þ ð1Þ þ4 a 1 tz4  R cos 0 R cos az4 q q 1

1

where at4 ¼[atx4,aty4,atz4]T and a4 ¼[ax4,ay4,az4]T is respectively the acceleration vector of the target and the missile in LOS coordinate system. The missile attitude equations with disturbances are considered as follows [25,26] 8 γ V cos θ > α_ ¼  mV Ycos β þ g cosV cos  ωx cos α tan β > β > > > > > þ ω sin α tan β þ ω þd > y z α > > <_ g sin γ V cos θ Z β ¼ mV þ V cos β þ ωx sin α þ ωy cos α þ dβ ð2Þ > > Y ð tan β þ sin γ V tan θÞ þ Z cos γ V tan θ >_ > γV ¼ > > mV > > > θ tan β α  ω sin α þ d > :  g cos γ V cos þ ωx cos y cos β γV V cos β   8 ω_ ¼ I x  Iy ωx ωy =Iz þ Mz =Iz þ dωz > < z ω_ y ¼ ðIz  Ix Þωz ωx =Iy þ My =Iy þ dωy   > :ω _ x ¼ I y I z ωy ωz =Ix þM x =Ix þ dωx

ð3Þ

where m is mass of missile; g is acceleration of gravity; Ix, Iy, Iz is rolling, yawing and pitching moments of inertia. V is velocity of missile; θ, ψV is flight path angle and heading angle; α, β, γV is

attack angle, sideslip angle and velocity bank angle, and |α| o90°, |β| o90°, |γV| o90°; ωx, ωy, ωz is body-axis rolling, yawing and pitching angular rate; ωx , ωy , ωz is the dimension- less form of ωx, ωy, ωz, ωx ¼ ωx L=V, ωy ¼ ωy L=V , ωz ¼ ωz L=V ; δx, δy, δz is equivalent aileron, rudder and elevator deflection; D, Y, Z is drag, lift and side force in velocity coordinate; Mx, My, Mz is rolling, yawing and pitching moment; dα, dβ, dγV, dωx, dωy, dωz are external disturbances. The equations of aerodynamic forces and moments are described by 2 3 δ β 2 3 cαD α þ cD β þ cδDx δx þ cDy δy þ cδDz δz D 6 7 δy 6 7 6 7 β δ δ ð4Þ 4 Y 5 ¼ Q S6 cαY α þ cY β þ cYx δx þ cY δy þ cYz δz 7 4 5 δ β δz δx y α Z cZ α þ cZ β þ cZ δx þ cZ δy þ cZ δz 2

Mz

2

3

δ

β

δz δx y z cαm α þ cm β þ cω m ωz þ c m δ x þ c m δ y þ c m δ z

6 ωy δy 6M 7 6 β δ δ 4 y 5 ¼ Q SL6 cαn α þ cn β þ cn ωy þ cnx δx þ cn δy þ cnz δz 4 δ β Mx cα α þ c β þ cωx ω þcδx δ þ c y δ þ cδz δ l

l

x

l

x

l

l

y

l

3 7 7 7 5

z

where S is reference area; L is reference length; Q¼ 0.5ρV2 is dynamic pressure; ρ is air density; cxy is partial derivative of aerodynamic force and aerodynamic moment coefficient for the corresponding variable, meaning of partial derivative y to x. 2.2. Design model The desired design model towards designing IGC law should be obtained by a series of transformation of the simulation model. In [10,11] the relationship between the LOS angles and the acceleration in the velocity coordinate is established by assuming that the missile flies heading to the target at initial time, so the missile velocity coordinate system approximately coincides with LOS coordinate system. But this cannot be satisfied in any cases. In order to break through this limitation, a novel relationship between LOS angles and acceleration components in the ballistic coordinate system is firstly established Transform a2 ¼ [ax2,ay2,az2] to O-xyz coordinate system and then to O-x4y4z4 coordinate system, one gets (6) 2 3 2 3 ax2 ax4         6a 7 7 1 ψV L1 θ 6 ð6Þ 4 y4 5 ¼ L q1 L q2 L 4 ay2 5 az4 az2         In (6), L q1 , L q2 , L  1 θ , L  1 ψ V is transfer 2 2 3 cos q2 cos q1 sin q1 0   6   6 7 L q1 ¼ 4  sin q1 cos q1 0 5; L q2 ¼ 4 0 sin q2 0 0 1 2

L

1

cos θ θ ¼6 4 sin θ 0

 

 sin θ

0

7 0 5; 1

cos θ 0

2

3 L

1



ψV



6 ¼4

matrix.

where 2 B0 ¼ 4

0

 R1

0

0

0

1 R cos q1

 sin q2

0

3

7 0 5 cos q2

1 0

cos ψ V

0

0

1

0

 sin ψ V

0

cos ψ V

Combining (1) and (6), one gets 2 3 " # ax2 q€ 1 6a 7 ¼ f Σ0 þ B0 4 y2 5 þ Δat0 q€ 2 az2

Fig. 1. Coordinate system between missile and target.

ð5Þ

sin ψ V

3 7 5

ð7Þ

3

        5L q 1 L q 2 L  1 ψ L  1 θ ; V

Please cite this article as: Cong Z, Yun-jie W. Non-singular terminal dynamic surface control based integrated guidance and control design and simulation. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.03.013i

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2 f Σ0 ¼ 4

_

 2RRq_ 1  q_ 2 2 sin q1 cos q1 _

 2RRq_ 2 þ2q_ 1 q_ 2 tan q1

3 5;

"

Δat0 ¼

aty4 R atz4 R cos q1

2

3 2 δz cα 6δ 7 QS 0 4 Y u ¼ 4 y 5; B1 ¼ m B0 cαZ

#

δx

2

The structure of matrix of B0 is 2  3, may wish to set its form as

"

B0 ¼

b11

b12

b13

b21

b22

b23

02

Assume ax2 ¼0 owing to q1 and q2 mainly depend on ay2 and az2, then 2 3 " # 0 ay2 6a 7 0 B0 4 y2 5 ¼ B0 az2 az2 " # b12 b13 where, B00 ¼ , Eq. (7) change as b22 23 " # " b# q€ 1 ay2 ð8Þ ¼ f Σ0 þB00 þ Δb0 þ Δat0 q€ 2 az2 where, Δb0 is the disturbance because of the assumption of ax2 ¼0. The acceleration components of the missile along the y-axis and z-axis in the ballistic coordinate system are given by ( Y Z ay2 ¼ m cos γ V  m sin γ V  g cos θ ð9Þ Y Z az2 ¼ m sin γ V þ m cos γ V It can be guaranteed γV E0 by the control of aileron δx, which will be integrated in the IGC law in next section. Furthermore gravity is ignored. Then (9) is changed to (10). " #   ay2 1 Y þ Δg γV ð10Þ ¼ az2 m Z

γV E0 and the

ð11Þ

where

2 3 δy δz δx Q S4 cL δx þ cL δy þcL δz 5 þ Δg γV Δa ¼ m cδx δx þ cδy δy þcδz δz Z

"

Z

Z

Then (8) can be rewritten as 2 3 # β " # α q€ 1 Q S 0 4 cL cL 5 α ¼ f Σ 0 þ B0 þ B00 Δa þ Δb0 þ Δat0 β β q€ 2 m cα c Z

ð12Þ

Z

In this way, a direct relationship between [α, β] and [q1,q2] is established by (12). Thus combining (2)–(5) and (12), the Integrated guidance and control design model can be obtained as a set of state equations as (13), which is a cascade system. 8 x_ 0 ¼ x1 ðaÞ > > > > < x_ 1 ¼ f 1 þ B1 x2 þ d1 ðbÞ ð13Þ x_ 2 ¼ f 2 þ B2 x3 þ d2 ðcÞ > > > > : x_ 3 ¼ f 3 þ B3 u þd3 ðdÞ

x0 ¼

q2

# ;

"

# q_ 1 x1 ¼ _ ; q2

"

#

α x2 ¼ ; β

2

α

3

6 7 x 2 ¼ 4 β 5;

γV

0 Q SLcδmz =Iz

2

1

6 B2 ¼ 4 0 0 3

6 7 δ 6 7 B3 ¼ 6 Q SLcny =Iy 7; 4 5 δx Q SLcl =Ix 2

β

cZ

"

5¼

b11

b12

b21

b22

# ;

sin α tan β

 cos α tan β

 sin αsec β

cos αsec β

cos α

sin α

3 7 5;

f 1 ¼ f Σ0 ;

 mV

g cos γ V cos θ Y þ V cos β cos β

3

6 7 6 7 g sin γ V cos θ Z 6 7 mV þ V cos β 6 7; f2 ¼ 6 "  #  7 6 Y tan β þ sin γ V tan θ þ 7 4 =ðmV Þ 5 Z cos γ V tan θ  mg cos γ V cos θ tan β 2

  3  ω β I x  I y ωx ωy =Iz þQ SL cαm α þ cm β þcmy ωz =Iz 6 7   6 7 ωy β 6 7 α f 3 ¼ 6 ðI z I x Þωz ωx =Iy þ Q SL cn α þ cn β þ cn ωy =Iy 7; 6 7    4 5 β x I y  I z ωy ωz =Ix þ Q SL cαl α þ cl β þcω ωx =Ix l d1 ¼ B00 Δa þ Δb0 þ Δat0 ; 2

3 dα 6d 7 6 d 7 d2 ¼ 4 22 5 ¼ 4 β 5; dγ V d23 d21

3

2

2  3 δ cδx δ þ cmy δy =Iz 2 3 6 m x dωz  7 6 7 6 d 7 6 cδx δ þcδz δ =I 7 6 d 7 d3 ¼ 4 32 5 ¼ 6 n x y 7 þ 4 ωy 5 n z 6  7 4 δy 5 d33 dωx δz cl δy þ cl δz =Ix 2

d31

3

The essence of integrated design philosophy is exposed during the transformation process from simulation model to design model. The direct affine relation between guidance equations and attitude equations is established by an ‘intermediate states’, and then the IGC law can be indeed designed for the obtained integrated cascade design model. In this paper, the ‘intermediate states’ are just aerodynamic angles α and β. The aerodynamic forces mainly come from aerodynamic angles, so the aerodynamic force Y and Z in (10) are substituted by (4). Then the guidance Eq. (10) controlled by forces is transformed to (12) which is controlled by aerodynamic angles. Exactly, the states of attitude equations are right aerodynamic angles. Thus the direct affine relation between guidance equations and attitude equations is established. The integrated cascade design model is obtained. Some physical limitations need to be considered, so the following assumptions are given.

T

where q1

3 b12 7 b22 5;

3

β

cY

Assumption 1.

T

"

b11 6 B1 ¼ 4 b21

#

where Δg γV is owing to the assumption that neglect of gravity. Combining (10) with (4), one can obtain 2 3 " # β " # α ay2 Q S4 cL cL 5 α ¼ þ Δa β az2 m cα cβ Z Z

3

2

ωz

3

6 7 x 3 ¼ 4 ω y 5;

ωx

(1) For uncertainty dynamics in system (13), ‖di ‖ rdi , di is a positive constant, i ¼ 1; 2; 3.     (2) There exists bounded set α ; α for attack angle and β ; β for sideslip angle due to promote  the  stability of missile. (3) There exists bounded set δ ; δ for deflection angles due to the physical limitations on actuators. 2.3. IGC law design objective The objective of IGC law can be elaborated as: design an IGC law that can guarantee the missile hits the target more accurately while better satisfying the terminal impact angular constraints. In other words, under the designed IGC law x0 approaches the expectations while x1, x2 and x3 trend to stability during the terminal guidance phase.

Please cite this article as: Cong Z, Yun-jie W. Non-singular terminal dynamic surface control based integrated guidance and control design and simulation. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.03.013i

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Remark 1. Terminal impact angular constraint generally refers to the constraints of trajectory angle θ-θc and ψV-ψVc at the end of the attack. The relationship between LOS angles and trajectory angle can be got in Ref. [27]. So in this paper, two LOS angles q1 and q2 are regarded as the angle constraints during the design procedure.

3. Design of integrated guidance and control law In this subsection, a novel NT_DSC IGC law is proposed. System (13) is a time-varying nonlinear cascade system with matched and unmatched uncertainties. For such kind of nonlinear system, the most natural control method is the back-stepping approach since the system satisfies the so called block low-triangular structure. But the traditional back-stepping methodology suffers from the problem of ‘explosion of complexity’ arising from the repeated differentiations of the virtual controls [10,11]. To avoid this problem, DSC methodology [24] is used for design of IGC law. On the other hand, NTSMC has the advantages of fast convergence and avoids the singularity problem of terminal sliding mode. So NTSMC is used under the framework of DSC to design the IGC law. In order to facilitate calculation, the following definition is given.

where κ1 ¼ diag(κ11,κ12), ξ ¼ diag(ξ11,ξ12), η ¼ diag(η11,η12), ε1 ¼ diag(ε11, ε12), κi ¼diag(κi1,κi2,κi3), ξi ¼diag(ξi1,ξi2,ξi3), ηi ¼ diag(ηi1, ηi2,ηi3), εi ¼diag(εi1,εi2,εi3), i ¼2,3. κij A R þ , ξij A R þ , ηij A R þ , εij A R þ . mij and nij are odd integers satisfying 0 omij/nij o1. As x0c and x1c are constants and x1c ¼0, combining (13)(a)(b), (14) and (17), the derivative of z1 can be rewritten as follows  p:=q  1  f 1 þ B1 x2c z_ 1 ¼ x1 þ p:=ðqbÞx1 h i   p:=q  1 m :=n  κ1 z1  ξ1 z1 1 1  p:=ðqbÞ η1 þ ε1 signðz1 Þ x1 one gets the virtual control vectors h     1 1 m :=n 2  p:=q x2c ¼ B1 b p:=q  κ1 z1  ξ1 z1 1 1  x1    η1 þ ε1 signðz1 Þ  f 1 Thanks to the turning way of skid-to-turn, the expectation of γV is 0, which is γVc ¼0. So we have h iT x2c ¼ αc β c γ Vc In order to prevent ‘explosion of complexity’, a low pass filter is used to obtain the approximate derivative of the command x2c, shown as

τ2 x_ 2d þ x2d ¼ x2c ; x2d ð0Þ ¼ x2c ð0Þ

Definition 1. Vector a ¼ ½a1 ; a2 ; ⋯; ai ; ⋯; an T and  T b ¼ b1 ; b2 ; ⋯; bi ; ⋯; bn

where τ2 is the filter time constant. Combining (13)(c), (15) and (18), the derivative of z2 can be rewritten as follows

Define a:m , a_ and a:=b as follows:  1 m2 mi mn T ; a:m ¼ am 1 ; a2 ; ⋯; ai ; ⋯; an

z_ 2 ¼ f 2 þ B2 x3c  x_ 2d     p:=q  1 m :=n ¼ x1  κ2 z2  ξ2 z2 2 2  p:=ðqbÞB1 z1  η2 þ ε2 signðz2 Þ

a_ ¼ ½a_ 1 ; a_ 2 ; ⋯; a_ i ; ⋯; a_ n T ;

one gets another virtual control vectors h    p:=q  1 m :=n  1 x1  κ2 z2  ξ2 z2 2 2  b p:=q BT1 z1 x3c ¼ B1 2    η2 þ ε2 signðz2 Þ  f 2 þ x_ 2d

 T a:=b ¼ a1 =b1 ; a2 =b2 ; ⋯; ai =bi ; ⋯; an =bn ; where i ¼1,2,…,n. 3.1. NT_DSC based IGC law

Obtain the approximate derivative of the command x3c through a low pass filter, shown as

Firstly, considering the non-linear system (13), a non-singular terminal dynamic surface [28,29] can be chosen as  T 1 z1 ¼ ðx0  x0c Þ þ b ðx1  x1c Þp:=q ¼ z11 z12 ð14Þ

where τ3 is the filter time constant. Combining (13)(d), (16) and (19), the derivative of z3 can be rewritten as follows

τ3 x_ 3d þ x3d ¼ x3c ; x3d ð0Þ ¼ x3c ð0Þ

This form of dynamic surface has a better convergence effect and takes into account the terminal impact angular constraint exactly right. And other two dynamic surfaces can be chosen as  T ð15Þ z2 ¼ x2  x2d ¼ z21 z22 z23  z3 ¼ x3  x3d ¼ z31

z32

z33

T

ð16Þ

þ

where bj AR , pj and qj are positive odd integers satisfying 1 opj/ qj o 2, j¼1,2. Adjusting the bj can satisfy the requirement of both the guidance precision and the angle constraint. Then, base on terminal attractor in [22,23], novel attractors can be obtained according to characteristics of NT_DSC by adding a p:=q  1 x1 term as follows: h i   p:=q  1 m :=n z_ 1 ¼ x1  κ1 z1  ξ1 z1 1 1  p:=ðqbÞ η1 þ ε1 signðz1 Þ ð17Þ   p:=q  1 m :=n z_ 2 ¼ x1  κ2 z2  ξ2 z2 2 2 p:=ðqbÞB1 z1    η2 þ ε2 signðz2 Þ 



  p:=q  1 m :=n z_ 3 ¼ x1  κ3 z3  ξ3 z3 3 3  η3 þ ε3 signðz3 Þ  B2 z2

ð18Þ ð19Þ

z_ 3 ¼ f 3 þ B3 u  x_ 3d     p:=q  1 m :=n ¼ x1  κ3 z3  ξ3 z3 3 3  η3 þ ε3 signðz3 Þ  B2 z2 one gets the final control vectors    h  p:=q  1 m :=n x1  κ3 z3  ξ3 z3 3 3  η3 þ ε3 signðz3 Þ u ¼ B1 3 i  BT2 z2  f 3 þ x_ 3d Finally, the NT_DSC IGC law is given as Algorithm 1 Algorithm 1.

8  T 1 > z1 ¼ ðx0  x0c Þþ b ðx1 x1c Þp:=q ¼ z11 z12 > > > > h    i h iT >   1 > 1 m :=n 2  p:=q > >  η1 þ ε1 signðz1 Þ  f 1 ¼ αc β c  κ1 z1  ξ1 z1 1 1 x1 > > x2c ¼ B1 b p:=q > > h iT > > > x ¼ αc β γ > ; γ Vc ¼ 0 > c Vc 2c > > > > > _ τ þx ¼ x ; x ð 0 Þ ¼ x2c ð0Þ x > 2 2c 2d 2d 2d <  T z2 ¼ x2  x2d ¼ z21 z22 z23 > > h    i >   > p:=q  1 m :=n  1 > > x3c ¼ B1 x1  κ2 z2  ξ2 z2 2 2  b p:=q BT1 z1  η2 þ ε2 signðz2 Þ f 2 þ x_ 2d 2 > > > > > > τ x_ þx3d ¼ x3c ; x3d ð0Þ ¼ x3c ð0Þ > > 3 3d >  T > > > z3 ¼ x3  x3d ¼ z31 z32 z33 > > h    i >  > p:=q  1 m3 :=n3 > 1 >  κ3 z3  ξ3 z3  η3 þ ε3 signðz3 Þ BT2 z2  f 3 þ x_ 3d : u ¼ B3 x 1

ð20Þ

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5

3.2. Stability proof of IGC closed-loop system Proof briefly the stability of the closed-loop system constituted by Eq. (13) and Eq. (20). Proof. Consider the Lyapunov candidate function 1 1 1 V ¼ zT1 z1 þ zT2 z2 þ zT3 z3 2 2 2

ð21Þ

Taking the derivative of (21) and making use of the virtual control law x2c, x3c and final law u gives V_ ¼ zT1 z_ 1 þzT2 z_ 2 þ zT3 z_ 3 h i    1 ¼ zT1 x_ 0 þ b p:=q x1 p:=q  1 x_ 1 þ zT2 f 2 þ B2 x3 þ d2  x_ 2d  þ zT3 f 3 þ B3 u þ d3  x_ 3d h   i  1 p:=q x1 p:=q  1 f 1 þ B1 z2 þ B1 x2d þ d1 ¼ zT1 x1 þ b   þ zT2 f 2 þ B2 z3 þ B2 x3d þ d2  x_ 2d þ zT3 f 3 þB3 u þ d3  x_ 3d h     p:=q  1 m :=n 1  zT1 κ1 z1  zT1 ξ1 z1 1 1 þ b p:=q zT1  η1 ¼ x1

Fig. 2. Phase locus.

þ ε1 Þsignðz1 Þ þB1 Δx2cd þ d1 Þ     p:=q  1 m :=n  zT2 κ2 z2  zT2 ξ2 z2 2 2  zT2 η2 þ ε2 signðz2 Þ þ x1     p:=q  1 m :=n  zT3 κ3 z3  zT3 ξ3 z3 3 3 þ zT2 B2 Δx3cd þ d2 þ x1    zT3 η3 þ ε3 signðz3 Þ þzT3 d3  2    

X p =q  1 m =n þ 1  κ 1j z21j  ξ1j z1j1j 1j þ pj = qj bj  η1j þ ε1j z1j

¼ x1jj j j¼1

þ z1j d~ 1j



3 X 3 h X

þ

i¼2j¼1

p =q  1 x1jj j



mij =nij þ 1 ij zij

 κ ij z2ij  ξ



 

i  ηij þ εij zij þ zij d~ ij where Δx2cd ¼ x2d  x2c , Δx3cd ¼ x3d  x3c , ‖Δx2cd ‖r Δ2 and ‖Δx3cd ‖ r Δ3 , Δ2 and Δ3 are unknown but bounded positive constant; d~ 1j indicates the jth element of B1 Δx2cd þ d1 , d~ 2j indicates the jth 1, d~ ij , the ijth element of B2 Δx3cd þd2 ; Combining Assumption





element of lumped disturbance, satisfies d~ r d^ where d^ is ij

ij

ij

unknown but bounded positive constant, i¼1,2,3, j¼ 1,2,3.   As 1 o pi =qi o 2, so 0 o pi =qi  1 o1. pi and qi are positive odd numbers, bi 40, then p =q  1 x1jj j 4 0 when x1j a0. Thus when x a 0, by setting up η 4 d^ , i ¼1,2,3, j¼ 1,2,3, it can 1j

ij

ij

guarantee V_ r 

2 X





λ1j z1j 

j¼1

3 X





λ2j z2j 

j¼1

3 X





λ3j z3j o 0

j¼1

pj =qj  1 40 ij x1j

when x1 a 0. At this time, Lyapunov where λij ¼ ε stability conditions are satisfied, the designed closed-loop system is asymptotically stable. By appropriately adjusting  κ ij and ξij , m =n þ 1

 κ ij z2ij  ξij zij ij ij can make the states fast approaching each sliding mode surfaces. When x1j ¼ 0, Consider the Lyapunov candidate function 1 1 V 0 ¼ zT2 z2 þ zT3 z3 2 2

ð22Þ

At this time, Lyapunov stability conditions are satisfied, x3c and u can guarantee the arrival of sliding mode surfaces z2 and z3. Plug x2c into (13)(b), it is obtained as follows   x_ 1 ¼ f 1 þ B1 x2c þ B1 z2 þ Δx2cd þd1     1  m :=n ¼ b p:=q  κ1 z1  ξ1 z1 1 1  η1 þ ε1 signðz1 Þ þ d~ 1 When z1j 4 0, x_ 1j r  ε1j and when z1j o 0, x_ 1j Z ε1j The phase locus is shown as Fig. 2. So when x1j ¼0, z1j will go to 0 in finite time. Then the designed closed-loop system is asymptotically stable. By appropriately adjusting m =n þ 1 can make the states fast approaching  κ ij and ξij ,  κ ij z2ij  ξij zij ij ij each sliding mode surfaces. 3.3. Transient dynamics arrangement and chattering reduction The initial errors of LOS angle at the beginning of terminal guidance may be most likely very large. It may cause system instability. Therefore, tracking differential (TD) [30] is given as follows, which is used in the IGC law to arrange the transient dynamics of expected instruction x0 and x1. ( x_ 0 ¼ x1  x_ 1 ¼  rsign x0  x0c þx1 jx1 j=ð2r Þ TD can make the dynamic tracking process fast and stable. In addition, chattering may appear in the system because the sign function is used to reject various disturbances in the design of closed-loop. A universal measure is using a smooth function instead of the sign function. A hyperbolic tangent function is used to substitute for the sign function in this paper. So sign function may be changed to tanh function as     sign zij - tanh kij zij where kij 4 0; i ¼ 1; 2; 3; j ¼ 1; 2; 3:

Taking the derivative of (22) and making use of the virtual control law x3c and final law u gives

The chattering can be reduced by tuning the parameters and kij properly.

0 V_ ¼ zT2 z_ 2 þzT3 z_ 3     ¼  zT2 η2 þ ϵ2 signðz2 Þ þzT2 B2 Δx3cd þ d2    zT3 η3 þ ϵ3 signðz3 Þ þ zT3 d3 3 X 3 h  

i X ¼  ηij þ ϵij zij þ zij d~ ij o 0

4. Simulation results and comparison

i¼2j¼1

In this section, the effectiveness and performance of the proposed NT_DSC IGC law is verified by numerical simulations. In

Please cite this article as: Cong Z, Yun-jie W. Non-singular terminal dynamic surface control based integrated guidance and control design and simulation. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.03.013i

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order to make comparisons, a traditional SMC and DSC based IGC law is designed as Algorithm 2. Algorithm 2. 8  T > z1 ¼ ðx1 x1c Þ þ Cðx0  x0c Þ ¼ z11 z12 > > > > iT >  h > > > x2c ¼ B1 1  k1 z1  η1 signðz1 Þ  f 1  Cx1 ¼ αc βc > > > > h iT > > > > x2c ¼ αc βc γ Vc ; γ Vc ¼ 0 > > > > > > τ2 x_ 2d þ x2d ¼ x2c ; x2d ð0Þ ¼ x2c ð0Þ > <  z z22 z23 T > z2 ¼ x2  xh2d ¼ 21 > i > > > > x3c ¼ B2 1  k2 z2  η2 signðz2 Þ  BT1 z1  f 2 þ x_ 2d > > > > > > τ3 x_ 3d þ x3d ¼ x3c ; x3d ð0Þ ¼ x3c ð0Þ > > >  T > > > z ¼ x3  x3d ¼ z31 z32 z33 > > 3 h i > > > 1 T > : u ¼ B3  k3 z3  η3 signðz3 Þ  B2 z2  f 3 þ x_ 3d xt ¼[xt,yt,zt]T, vt ¼[vtx,vty,vtz]T respectively represents the position and velocity vector of the target. xm ¼[xm,ym,zm]T and vm ¼[vmx, vmy,vmz]T is respectively the position and velocity vector of the missile in the inertial coordinate system. Define  T  T xr ¼ xr ; yr ; zr ¼ xt  xm ¼ xt  xm ; yt ym ; zt  zm and  T vr ¼ ½vr ; vr ; vr T ¼ vt  vm ¼ vtx  vmx ; vty vmy ; vtz  vmz Then, the states that R, q1 and q2 are calculated by 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > R ¼ x2r þy2r þz2r > > h pffiffiffiffiffiffiffiffiffiffiffiffiffiffii < q1 ¼ arctan yr = x2r þ z2r > >  > : q2 ¼  arctan zr =xr The relative angular rate can be obtained by getting the time derivative of q1 and q2 8 v x2 þ z2  yr ðxr vx þ zr vz Þ > ffi < q_ 1 ¼ y ð r r2Þpffiffiffiffiffiffiffiffiffiffi 2 2 xr þ zr

R

zr vx > : q_ 2 ¼  vzxx2r  þ z2 r

r

For all numerical simulations presented in this section, the parameters and initial states are provided in Table 1. The limitations are  12 3 r α r 12 3 ,  12 3 r β r 12 3 , and  20 3 r δi r 20 3 , i ¼ x; y; z.

In simulation study, the following cases whose differences are caused mainly by the different lumped disturbances and the different initial locations of target. Case 1. Suppose that the coefficients of aerodynamics forces and moments are all reduced by 20% of their respective nominal values. The external disturbance comes from a gust wind whose speed is [20þsin(t)]m/s which is equivalent to the 8 grade wind. The gust wind blows to the missile by two angles θw ¼  30°and ψw ¼  60°. Here θw is the angle between the wind direction and the O-xz plane of the inertial frame, the upward is positive. And ψw is the angle between the wind direction and the O-x axis of the inertial frame, the right is positive. The initial position of the target is [xt0,yt0,zt0] ¼[ 0,0,0]m. Case 2. This case serves for the Monte-Carlo simulations. Suppose that the coefficients of aerodynamics forces and moments all offset randomly by  20%  20% of their respective nominal values. Moreover, the external disturbance comes from a gust wind whose speed is randomly  20þ sin(t)  20 þsin(t) m/s. The wind direction is expressed by two random angles θw ¼  30°  30° and ψw ¼  60°  60°. The initial position of the target is randomly [xt0, yt0,zt0] ¼[  1000  1000,0,  1000  1000] m. Remark 3. It is noted that system (13) is only used for IGC design, but not for the 6-DOF nonlinear simulations. The original nonlinear motion model of the missile with 15 states given in [31] is adopted for the 6DOF nonlinear simulations. And the aerodynamic parameters of forces and moments come from the Ref. [11]. Remark 4. The disturbance caused by the gust wind is equivalent to the attack angle and the sideslip angle which is explained in detail in [32]. Remark 5. TD is used in all simulations. The 2 IGC methods will simulate in the 2 cases above. The time histories of LOS angles (x0), LOS rates (x1), aerodynamic angles (x2), angular rates (x3), deflection angles (u), locations (xyz), velocity (V) and angles in ballistic coordinate system (θ,ψV) are presented in all simulations. Simulations start from Case 1. Simulations of the 2 methods mentioned in Table 1 have been carried out under the given initial states. The expected constraints are θc ¼  80° and ψVc ¼ 0°.The comparisons are shown in Figs. 3–9. It can be generally observed that the missile hits the target with certain accuracy and the ballistic is smooth shown as Fig. 3. The proposed NT_DSC IGC law presents better performance than

Table 1 Initial states and simulation parameters.

NT_DSC

SMC_DSC

TD

x0(0) ¼[  20.96,5.28]deg;x1(0) ¼[  0.96,0.26]deg/s; x2(0) ¼[3.9,3.5,0]deg; x3(0)¼ [0, 0, 0]deg/s; U(0)¼ [0, 0, 0]deg; [xm(0),ym(0),zm(0)] ¼[  38000,14500,3500]m [V(0),θ(0),ψ V (0)] ¼[1900 m/s,7.845°,  3.544°] b ¼ diag[0.01,0.01]; κ1 ¼ diag[0.005, 0.0025]; ξ1 ¼ diag[0.01, 0.01]; η1 ¼ diag[0.005, 0.005]; ɛ1 ¼diag[0.001 0.001]; p ¼[5, 5]T; q ¼ [3, 3]T;m1 ¼ [3, 1]T; n1 ¼ [5, 3]T; κ2 ¼ diag[10, 5, 5]; ξ1 ¼diag[0.1, 0.1, 0.1]; η2 ¼ diag[0.5 0.5 0.5]; ɛ2 ¼diag[0.001,0.001,0.001]; m2 ¼ [3, 1]T; n2 ¼ [5, 3]T; κ3 ¼ diag[50, 30, 30]; ξ3 ¼ diag[0.001,0.001,0.001]; η3 ¼ diag[2, 2, 2]; ɛ3 ¼ diag[0.001,0.001,0.001]; m3 ¼ [3, 1]T; n3 ¼ [5, 3]T; C ¼ diag[0.06, 0.04]; k1 ¼ diag[0.12, 0.12]; k2 ¼ diag[5, 5, 1]; k3 ¼ diag[10, 5, 7.5]; η1 ¼diag[0.03, 0.03]; η2 ¼ diag[0.5, 0.5, 0.5]; η3 ¼ diag[2, 1, 1.5]; r ¼ 20

x 10

NT__DSC SMC__DSC

4

2

1.5

y(m)

Initial states

1

0.5

0 3k

0

0

-0.5

-1

-1.5 x(m)

-2

-2.5

-3

-3.5 x 10

4

z(m) Fig. 3. Three-dimensional trajectory.

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-20

0

z

q1(°)

-40

NT__DSC SMC__DSC

0.2

(rad/s)

NT__DSC SMC__DSC

-0.2 -60

0

5

10

15

20

25

30

35

NT__DSC SMC__DSC

10

15

20 t(s)

25

30

35

40

NT__DSC SMC__DSC

0.2 0 -0.2

0

5

10

15

4

20 t(s)

25

30

40

0

x

0

35 NT__DSC SMC__DSC

0.2

2

(rad/s)

q 2(°)

6

5

y

t(s)

(rad/s)

-80

0

0

5

10

15

20 t(s)

25

30

35

40

-0.2

0

5

10

15

Fig. 4. LOS angles (x0).

20 t(s)

25

30

35

40

Fig. 7. Angular rates (x3).

-1

-2

-3

15 10 5 0 -5

z(°)

dq1(°/s)

0

NT__DSC SMC__DSC 0

5

10

15

20

25

30

y(°)

0.5

NT__DSC SMC__DSC

5

-2

15

20 t(s)

25

30

x(°) 0

5

10

15

20

25

30

35

0

5

10

15

20 t(s)

25

30

-0.5

0

5

v(m/s)

NT__DSC SMC__DSC 10

15

20 t(s)

25

30

35

20 t(s)

25

30

35

40

NT__DSC SMC__DSC

1000

40

0

NT__DSC SMC__DSC

5

10

15

20 t(s)

25

30

35

40

50 NT__DSC SMC__DSC

0 -50

0

5

10

15

20 t(s)

25

30

35

40

0

5

10

15

20 t(s)

25

30

(°)

20

0

V

(°)

15

1500

2 V

40

Fig. 8. Deflection angles (u).

(°)

(°)

4 2 0 -2 -4

10

2000

5

35 NT__DSC SMC__DSC

Fig. 5. LOS rates (x1).

0

40

0

t(s)

5 0 -5 -10 -15

35 NT__DSC SMC__DSC

0.5 -0.5

-1

(°)

10

0

0

dq2(°/s)

0

2

35

t(s)

NT__DSC SMC__DSC

-2 0

5

10

15

20 t(s)

25

NT__DSC SMC__DSC 30 35 40

35

40

NT__DSC SMC__DSC

10 0 0

5

10

15

20 t(s)

25

30

35

40

Fig. 6. Aerodynamic angles (x2).

Fig. 9. Velocity, flight path angle and heading angle.

SMC_DSC method. NT_DSC IGC law can guarantee the missile hits the target with higher accuracy while better satisfying the terminal impact angular constraint. Moreover, as Remark 1 in Section 2.3, θ and ψV tend to the terminal impact angular constraints when LOS angles (x0) tends to the constraints shown as Fig. 10. In addition, the states (x2 and x3) and commands (u) are reasonable

and smooth without any chattering. Concrete accuracy statistics will be presented by Monte Carlo simulations as follows. In order to investigate and research the 2 methods more generally, Monte-Carlo simulations in Case 2 continue. The miss distances and terminal impact angular constraints are concerned in the following simulations.

Please cite this article as: Cong Z, Yun-jie W. Non-singular terminal dynamic surface control based integrated guidance and control design and simulation. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.03.013i

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1) The essence of integrated design philosophy is summarized: establish the direct relation between guidance and attitude equations by an ‘intermediate state’, and then design an IGC law for the obtained integrated cascade design model. 2) The superiority of the proposed NT_DSC IGC law mainly thanks to the non-singular terminal dynamic surface chosen as (14) and terminal attractor based novel attractors chosen as (17)– (19). Dynamic surface as (14) has a better convergence effect by adjusting pj and qj conveniently and can take into account the terminal impact angular constraint exactly right by adjusting the bj. Attractors as (17)–(19) can resist the lumped disturbance and guarantee the strict stability of the closed-loop. Simulations of comparisons with SMC_DSC IGC law are effectively support this point.

Acknowledgment Fig. 10. Miss distances.

This research has been funded in part by the National Natural Science Foundation of China under Grant 91216304. The authors would also like to thank the viewers and the editor for their comments and suggestions that helped to improve the paper significantly.

-65

-70

(°)

NT-DSC SMC-DSC

-75

References -80

0

5

10

15

20

25 30 N(times)

35

25 30 N(times)

35

40

45

50

NT-DSC SMC-DSC

6

V

(°)

4 2 0 -2

0

5

10

15

20

40

45

50

Fig. 11. Terminal impact angular constraints.

Table 2 Statistical results of Monte-Carlo simulations.

μMiss(m) σMiss(m) μ(θ)deg σ(θ)deg μ(ψV)deg σ(ψV)deg

NT_DSC

SMC_DSC

1.6070 0.3325  78.5406 0.0701 0.0013 0.0045

2.4852 0.8479  66.5222 0.0736 3.2756 0.1668

50 times simulations of the 2 IGC laws mentioned in Table 1 have been carried out in Case 2. The miss distances are shown as Fig. 10. And the terminal impact angular constraints are shown as Fig. 11. Statistical results of Monte Carlo simulations are given as Table 2 to observe and analyze the expectations (μ) and normalized variances (σ) of the miss distances and angular constraints. Obviously the miss distances of NT_DSC are less. And it exhibits much better performance in the aspect of satisfying the angular constraints. 5. Conclusion In this paper, a novel NT_DSC IGC law is developed to address the terminal guidance problem.

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Please cite this article as: Cong Z, Yun-jie W. Non-singular terminal dynamic surface control based integrated guidance and control design and simulation. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.03.013i