Flight Controller Design for Intercepting Missiles with ... - IEEE Xplore

Proceedings of the 2007 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 2007

FrB10.5

Flight Controller Design for Intercepting Missiles with Multiple TVC Systems and DCS I-Fan Lin, Chin-I Huang, and Li-Chen Fu, Fellow, IEEE

Abstract— In this paper, we propose a highly maneuverable autopilot system based on multiple Thrust Vector Control (TVC) mechanisms and Divert Control System (DCS). The strategy of the cooperation of multiple TVC mechanisms and DCS is discussed. Moreover, the decision and control part in missiles: guidance law (GL) and autopilot is presented. The GL is designed with dynamic sliding mode control (DSMC) to eliminate the chattering phenomenon caused by sliding mode control (SMC) and to minimize the distance between the missile and the target without the estimation of interception time. The autopilot controller based on quaternion representation is designed using backstepping control technique to execute the attitude command. The stability of the integrated guidance/autopilot (G/A) system is analyzed by Lyapunov stability theory. In addition, we advocate a wingless missile to reduce the nonlinear effect from the aerodynamics as much as possible. Extensive simulations including aerodynamic model are finally demonstrated to verify the validity of the proposed integrated G/A systems of missiles incorporating the highly maneuverable inputs.

T

I. INTRODUCTION

HE control of flight vehicles are widely applied from the earth to the universe, such as airplane’s cruising flight, trajectory tracking of launch vehicles, and target interception of missiles. Our principal objective in this paper is to design a robust controller for missiles in order to ensure high performance in translational and rotational motions regardless of the possible external disturbances and internal perturbations. Most of the guidance laws (GLs) are derived from the proportional navigation (PN) GL and the optimal guidance law [1], where the objective of the previous one is to maintain the direction of the line of sight (LOS) due to its ease of implementation and efficiency. Besides PNGL, many other formulations on the guidance laws are also presented such as the 3D modeling method [2], game theory [3], fuzzy-logic method [4], neural networks control method [5], and sliding-mode guidance [6]. Most GLs, including PN and its modifications, are developed to reduce the miss distance [7] and time of capture [8]. Moreover, attitude control is an important issue to be addressed for successful missile’s operation. In selection of

I-Fan Lin and Chin-I Huang are with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C.; e-mail: [email protected] Li-Chen Fu is with the College of Electrical Engineering and Computer Science, National Taiwan University, Taipei, Taiwan, R.O.C.; e-mail: [email protected]

1-4244-0989-6/07/$25.00 ©2007 IEEE.

appropriate coordinate systems, quaternion representation is often used in navigation and guidance algorithm to eliminate the singularities present in direction sine matrices [9]. On the other hand, a missile equipped with TVC mechanism can control its acceleration direction effectively [10] when the missile’s fin fails, which in turn implies that the maneuverability or controllability of the missile can be enhanced during the stage while the speed of the missile is slow or the air density surrounding the missile is low. In this paper, the missile’s interception problem is addressed, either against offensive missiles or enemy air fighters. In this case, two main systems are the main topics of this paper: guidance laws and autopilot. GLs guide an airframe directions based on the current attitude; autopilot system depends on a well-designed controller to follow the guidance commands stably. Furthermore, the DSMC GL can eliminate the chattering phenomenon that always occurs when using SMC. In order to enhance the robustness of the controller, we propose a backstepping control technology with variable structure control idea for the controller design of nonlinear integrated G/A system with highly maneuverable actuators, consisting of multiple TVC systems and DCS. Various simulations including aerodynamic model are demonstrated to verify the feasibility and test the performance of the missile’s integrated G/A systems. Moreover, the comparison between the DSMC GL and SMC GL is also discussed in the simulation section. II. MISSILE MODEL In this paper, the physical structure of the designed airframe is shown in Figure 1. We simplify the airframe model as a cylinder (wingless airframe) in order to reduce the effect from the aerodynamics as much as possible. With the view of this kind of wingless airframe, the aerodynamics will be minute and can be regarded as disturbances, which renders the control surfaces useless. Hence, we can focus ourselves on the whole-phase-single-strategy controller design. The airframe structure includes the main TVC mechanism, extra TVC mechanism, and DCS, where TVC is a mechanism realized by a moveable tail nozzle, and DCS is an additional auxiliary propellant system mounted on the lateral of the missile. Apparently, when the aerodynamics fails in space, the lateral force and torque are generated to impose a large impact on the missile. For survival under such uncontrollable aerodynamics, TVC and DCS that provide their superior maneuverability are widely adopted [11]. On account of the current mechanical design capability, the maximum moving

5248

FrB10.5

angle of TVC is restricted with 15 degrees, and the maximum propellant power of DCS is far smaller than the main thrust. Therefore, we add four TVC mechanisms to compensate for the loss of the torque produced by the main TVC system. x N

y DCS

CG

extraTVC

lDCS

mainTVC

r

θ

N

θ

x

lTVC

z φ

l

y

z

Fig. 1. (a) Missile configuration (b) Main TVC moving angles Moreover, Figure 2(a) shows four extra symmetrical TVC mechanisms, two mounted parallel to x − y plane and two to x − z plane, respectively. We assume that the upper and the lower extra TVC mechanisms are identical, i.e., they have the same moving angle θ ul . Besides, they can just move in the x − y plane. Similarly, the left extra TVC mechanism is on a par with the right one, i.e., having the same moving angle θ lr and they can just move in the x − z plane. The moving angles of the four extra TVC mechanisms are shown in Figure 2(b). y x Ne z

Ne

θul

θlr

z

y

Fig. 2. (a) Extra TVC

(b) Extra TVC moving angles

Physically obtained, the kinematic equations of the missile are as follows [12]. Translations: Fb _ prop = Fb _ mainTVC + Fb _ DCS + Fb _ extra ⎡ N cos θ + 2 N e (cos θ ul + cos θ lr ) ⎤ = ⎢ N sin θ cos φ + Fb _ DCS _ y + 2 N e sin θ ul ⎥ , (1) ⎣⎢ N sin θ sin φ + Fb _ DCS _ z + 2 N e sin θ lr ⎦⎥ where N is the main thrust ,and N e is the extra thrust. Rotations: Tb _ prop = Tb _ mainTVC + Tb _ DCS + Tb _ extra 0 ⎡ ⎤ (2) = ⎢ NlTVC sin θ sin φ − l DCS Fb _ DCS _ z + 2 N e lTVC sin θ lr ⎥ . ⎢⎣− NlTVC sin θ cos φ + l DCS Fb _ DCS _ y − 2 N e lTVC sin θ ul⎥⎦

follows: (3) a = v =  r =  rTarget −  rM issile , and rMissile are the acceleration generation from

where rTarget their own thrust regardless of gravity. In this paper, we assume the earth is flat and gravity is constant and both the missile and the target are affected by the same gravity [13]. Note that a , v , and r (shown in Figure 3) are generally not on the same plane in 3-D space. According to (3), we can get d r = d r2 + r2 + r2 = v ⋅r r (4) 2 3 dt dt 1 and define rˆ = r r , then

d rˆ = d (r r ) = (1 r ) v − ( v ⋅ rˆ r ) rˆ . (5) dt dt Let v & is the projection of v on r and v ⊥ = v − v & . Based on the triangular equations, we have: (6) v & = ( v ⋅ rˆ )rˆ , v ⊥ = v − ( v ⋅ rˆ ) rˆ and define a ⊥ = a − ( a ⋅ rˆ ) rˆ , then tacking the time derivative

of (6), we get: v ⊥ = v − ⎡( v ⋅rˆ ) rˆ + ( v ⋅rˆ ) rˆ + ( v ⋅rˆ ) rˆ ⎤ ⎣ ⎦ 2 ˆ 2 = a − ⎡(a ⋅ rˆ )rˆ + ( vr⋅v − ( v⋅rr ) )rˆ + vr⋅rˆ v − ( v⋅rr ) rˆ ⎤ ⎣ ⎦ 2 v⊥ v⋅rˆ = a ⊥ − r rˆ − r v ⊥ .

Note that a ⊥ and v ⊥ are not in the same direction and v ⊥ ⋅ rˆ = 0 . If v ⊥ → 0 along with r → 0 , the missile will hit the target. In other words, if v ⊥ is always equal to zero and r is decreasing, which means that v always points to the opposite direction of r , the missile and the target will meet eventually. As such, our goal is to design a ⊥ to reduce v ⊥ . If a Target is acquirable, it is easy to dig out the relationship a ⊥ = a T ⊥ − a M ⊥ , and if a Target is merely constant gravity, then a M ⊥ = − a ⊥ . The main source of accelerating propulsion comes from main thrust engine TVC propellant N and the total mass of the missile is m . As a result of this, we have 2

2

(8)

aM = N m = aM ⊥ + aM & .

Recall that a M ⊥ = a M − ( a M ⋅ rˆ ) rˆ and a M & = (a M ⋅ rˆ )rˆ , we obtain : 2 2 (9) a M & = ( N m ) − a M ⊥ rˆ .

)

(

Consequently, the desired acceleration of the missile a d would be: 2 2 a d = a M ⊥ + a M & = a T ⊥ − a ⊥ + ( Nm ) − a T ⊥ − a ⊥ rˆ , (10)

)

(

which is delivered to the autopilot system.

III. CONTROL DESIGN

(7)

v

a

In this section, the missile’s controller design is divided into two parts: guidance system and autopilot system. A. Guidance System In this subsection, the general missile intercepting problem is discussed. Define the relative position, velocity and acceleration to be the vector from the missile to the target as 5249

a⊥

a&

v⊥

Target v&

r

Missile

Fig. 3. Missile-Target Engagement in 3D space

FrB10.5

Now, according to (7), we propose dynamic sliding mode control methods to design a ⊥ comparing with sliding mode control [14] in the simulation. Dynamic Sliding Mode Control

In traditional, the trajectory reverses its direction and heads again toward the surface by SMC. Once again it crosses the surface and repetition of this process creates the oscillation known as chattering that is easy to lead to low control accuracy. The chattering may also excite unmodeled high-frequency dynamics, which degrades the performance of the system and may even give rise to instability. To alleviate the problem with chattering, we would like to use DSMC to reduce the chattering phenomenon [15]. The concept of the DSMC is that the sliding surface not only depends on the state of the system but also on the system’s inputs. Now, consider the system 2 ⎧⎪ v ⊥ = − v ⊥ rˆ − v ⋅rˆ v ⊥ + u r r (11) , ⎨ ⎪⎩ u = h where h is our new control input (pseudo control input). Define the sliding surface variable as (12) S g 2 = C g 2 v ⊥ + u, where C g 2 is a 3 × 3 diagonal P.D. matrix. Next, taking V ( S g 2 ) = 12 S Tg 2 S g 2 as a Lyapunov function candidate, we can get the control law as h = Cg2(

v⊥ r

2

rˆ +

v ⋅rˆ r

(13)

v ⊥ − u ) − λ g 2 sgn( S g 2 ),

theory [16], the equilibrium point v ⊥ = 0 is globally asymptotically stable. B. Autopilot System The attitude of a rigid body is expressed by quaternion q and angular velocity ω . (q e , ω e ) are defined as the errors between desired values (q d , ω d ) and current values (q , ω ) as follows: ⎡ q e1 ⎤ ⎡ q d 4 q d 3 − q d 2 − q d 1 ⎤ ⎡ q1 ⎤ ⎢ q e 2 ⎥ = ⎢ − q d 3 q d 4 q d 1 − q d 2 ⎥ ⎢ q 2 ⎥ , ω = ω − ω , (16) e d ⎢ q e 3 ⎥ ⎢ q d 2 − q d 1 q d 4 − q d 3 ⎥ ⎢ q3 ⎥ ⎢⎣ q e 4 ⎥⎦ ⎢⎣ q d 1 q d 2 q d 3 q d 4 ⎥⎦ ⎢⎣ q 4 ⎥⎦ where the quaternion is defined as: T

T (17) q = [ q q 4 ] = ⎡⎣ eˆ sin( φ2 ) cos( φ2 ) ⎤⎦ , where the unit vector eˆ = [ e1 e2 e3 ]T and angle φ are the

Euler Axis and Angle. In Figure 4, as the desired acceleration commands a d are generated from GLs, it can be seen that the missile’s nose direction xb should be aligned with the vector a d . Referring to [13], the vector eˆ and the angle φ can be obtained by the following equations: eˆ = xi × xb = xi × a d , φ = cos − 1( xi ⋅ xb ) = cos − 1( xi ⋅ a d ), (18) xi × xb xi × a d xi xb xi a d which in turn yield the desired angular velocity by the kinematic equation of the quaternion: ⎡ qd 4 qd 3 − qd 2 − qd 1 ⎤ (19) ω d = 2 ⎢− qd 3 qd 4 q d 1 − q d 2 ⎥ q d . ⎣⎢ q d 2 − q d 1 q d 4 − q d 3 ⎦⎥

where λ g 2 is a scalar and h is a discontinuous signal. Then, we can obtain the actual control u = ∫ h dt , and u is continuous, so that the chattering will be eliminated. Proof : Take the derivative of V (S g 2 ) , we get V ( S g 2 ) = S Tg 2 (C g 2 v ⊥ + u ) = − λ g 2 S Tg 2 sgn( S g 2 ) (14) = − λ g 2 S g 2 1 ≤ 0. According to the Lyapunov theorem, it can be shown that the system trajectory will reach the surface S g 2 = 0 in finite time. From the sliding mode theory, once the reaching condition is satisfied, the system is eventually forced to stay on the sliding surface, i.e., S g 2 = C g 2 v ⊥ + u = 0. Now we verify whether v ⊥ and u will converge to zero when system trajectories are on the sliding surface. Taking V ( v ⊥ ) = 12 v T⊥ v ⊥ as a Lyapunov function candidate and u = − C g 2 v ⊥ , we obtain

(

V ( v ⊥ ) = v T⊥ v ⊥ = v T⊥ ( − ≤−

v ⋅rˆ r

v ⋅rˆ r

I 3× 3 − C g 2 ) v ⊥ −

v v ⊥ − λ min ( C g 2 ) v v ⊥ , T ⊥

T ⊥

v⊥ r

2

rˆ

)

(15)

where we use the fact that v T⊥ rˆ = 0 , and I 3× 3 is a 3 × 3 identity matrix, and λ min ( C g 2 ) is the smallest of C g 2 . If we choose all entries in C g 2 large enough, then V ( v ⊥ ) is negative definite (N.D.). According to the Lyapunov stability

zi

zb

eˆ

yb

zb

yb

ad

xb

xb

φ

yi

xi

Fig. 4. Desire Quaternion After transforming the guidance commands to the autopilot, we propose backstepping control method to design the autopilot controller. The backingstepping technique is a systematic and recursive design methodology for nonlinear feedback control [17]. It breaks a design problem for full system into a sequence of design problems for lower-order systems. In addition, the backstepping control method guarantees asymptotic stability [18]. Applying the kinematic equation of the error quaternion and the Euler’s equation ⎧ q e = 12 q e × ω e + 12 q e 4 ω e (20) ⎨ ω − ω × ( I ω ) + Tb .  I I ω = − ⎩ Replace ω = ω e + ω d , then we have dynamic equations as

5250

FrB10.5

follow: ⎧ q e = 12 q e × ω e + 12 q e 4 ω e ⎪ ⎨ I ω e = − I ω d − Iω e − Iω d − ω e × ( I ω e ) ⎪⎩ − ω e × ( I ω d ) − ω d × ( I ω e ) − ω d × ( I ω d ) + Tb , (21) where Tb is the control. We start with the first equation of (21) with ω e being viewed as the input, and then proceed to design the feedback control ω e = φ ( q e ) to stabilize the origin q e = 0 . With ω e = φ ( q e ) = − Pq e , where P is a P.D., the Lyapunov function candidate 3× 3 V1 ( q e ) = 12 q Te q e satisfies (22) V1 ( q e ) = q e ⋅ ( q e × ω e ) + 1 q e 4 q Te ω e = − 1 q e 4 q Te Pq e , 2

2

where q e 4 = − 12 q e ⋅ ω e = 12 q Te Pq e ≥ 0, which implies that q e 4 is a non-decreasing positive variable. In other words, 0 ≤ q e 4 ( t 0 ) ≤ q e 4 ( t ) ≤ 1, ∀ t ≥ 0 . Substituting this relation into (22), we can get (23) − 12 q Te Pq e ≤ V1 ( q e ) ≤ − 12 q e 4 ( t 0 ) q Te Pq e ≤ 0. If we choose a nonzero q e 4 ( t 0 ) , namely, the x -axes of the inertial frame as well as the body frame are not opposite to each other, the rotation angle φ will not be equal to ± π , and thus q e 4 ( t 0 ) will remain bounded away from zero. As a result of this, we have V1 ( q e ) is P.D., and V1 ( q e ) is N.D. Additionally, V1 ( q e ) → ∞ as q e → ∞ . According to the Lyapunov stability theory [16], the equilibrium point q e = 0 is globally asymptotically stable. However, we should beware that ω e can not be directly assigned such that ω e = φ ( q e ) since it is really Tb that is the control. To be practical, we use the change of variables, namely, z 2 = ω e − φ ( q e ) = ω e + Pq e to transform the system into the form ⎧ q e = 12 q e × ( z 2 − Pq e ) + 12 q e 4 ( z 2 − Pq e ) ⎪ ⎨ I z 2 = − I ω d − Iω e − Iω d − ω e × ( I ω e ) − ω e × ( I ω d ) (24) ⎪⎩ − ω d × ( I ω e ) − ω d × ( I ω d ) + Tb + I Pq e . T T 1 1 Now, we take V 2 ( q e , z 2 ) = 2 q e q e + 2 z 2 I z 2 as a Lyapunov function candidate. Then differentiating it (disturbances are taken into consideration) to yield V2 = 12 q e 4 q Te z 2 − 12 q e 4 q Te Pq e + z T2 ( Tb + I Pq e + d a ) (25) + z T2 ( 12 Iz 2 − I ω d − Iω − ω × ( I ω )), where d a is the summation of disturbances that come from parametric uncertainties as well as unmodeled dynamics. Assuming d a ≤ d a _ max , we can desire the control law as: Tb = I ω d + Iω + ω × ( I ω ) − I Pq e − 12 Iz 2 − 12 q e 4 q e − ( d a _ max + η a ) sgn( z 2 ),

(26)

where η a (3×1) > 0 is an adjustable vector variable. With the designed control law, we have V2 = − 12 q e 4 q Te Pq e + z T2 d a − z T2 d a _ max sgn( z 2 ) − z T2 η a sgn( z 2 ) (27) ≤ − 12 q e 4 (t 0 ) q Te Pq e − ηTa z 2 ≤ 0. 1

Accordingly, V2 ( q e , z 2 ) is N.D. and then the origin is globally asymptotically stable. C. Integrated Guidance/Autopilot Systems We assume that we have a perfect autopilot system to execute the guidance command and thus achieve the overall task while deriving the guidance law. That assumption, however, cannot be true in the real world. As a result, we are supposed to re-derive the stability analysis for the integrated G/A system with an imperfect autopilot system. For the imperfect autopilot system, the actual acceleration should be given a c rather than a d . The relationship between them can be found in [12], and the final result is: (28) a c ⊥ = (1 + E ) a d ⊥ . If we let P = p I 3× 3 in z 2 = ω e + Pq e , and the control a d ⊥ ≠ 0 , we can find the bound of E satisfying E ≤

1 p2

E A , where E A is a bounded positive scalar.

Next, we analyze two integrated G/A system. The first is guidance system using SMC and the other is that using DSMC. a)

Guidance with Sliding Mode Control

Define the Lyapunov function candidate of the integrated G/A system as (29) V = 12 S Tg 1S g 1 + 12 q Te q e + 12 z T2 I z 2 , Substitute a c ⊥ for a d ⊥ in (28), we can find [12] that V ≤ − λ g 1 S g 1 1 + λ max ( C g 1 ) E

v ⋅rˆ r

v⊥

2

− 12 q e 4 ( t 0 ) q Te Pq e − ηTa z 2 1 .

(30)

where S g 1 = C g 1 v ⊥ and C g 1 is also a 3 × 3 diagonal P.D. matrix. Recall that v always points at the opposite direction of r , so that v ⋅ rˆ < 0 . According to the Lyapunov direct method, we can conclude that the overall integrated G/A system is globally asymptotically stable in the presence of the attitude error. b)

Guidance with Dynamic Sliding Mode Control

Define the Lyapunov function candidate of the integrated G/A system as (31) V = 12 S Tg 2 S g 2 + 12 q Te q e + 12 z T2 I z 2 . In the same way, we can find [12] that 2 V ≤ − λ g 2 (1 + E ) S g 2 1 + λ max ( C g 2 ) E vr⋅rˆ v ⊥ (32) 2 v + E a T⊥ C g 2 ⊥r rˆ − 12 q e 4 ( t 0 ) q Te Pq e − ηTa z 2 1 . If we choose a large enough λ g 2 , V is N.D. According to the Lyapunov direct method, we can conclude that the overall integrated G/A system with the suggested system parameters is globally asymptotically stable. IV. ACTUATORS COOPERATION STRATEGY After computing the torque commands, the autopilot system controller transfers them to the actuators. Then, how to allocate those commands to multiple TVC systems and

5251

FrB10.5

DCS is momentous and significant. We have two assumptions here. One is that DCS are placed further to the front of the missile for the purposes of high maneuverability. Furthermore, DCS can easily eliminate the unpleasantly accompanied lateral force generated by the TVC system. Under this assumption, there is no coupled power exerted except for the constant propulsion from the main propellant engine along the body axis. In other words, this assumption ensures a rotational torque without lateral force accompanied, which is a proper way for the missile to change the direction of heading. The other is that the four extra TVC systems do not start until the magnitude of the moving angle of the main TVC system, θ , exceeds 15 degrees. That is to say, we only use the main TVC system and DCS in normal condition unless the main TVC system cannot reach the command torque calculated by the autopilot system controller. And then, we start the four extra TVC systems to compensate for the loss torque. Once the moving angle of the main TVC system gets back within the range of 15 degrees, the four extra TVC systems will immediately be turned off. To do so, we simply let the summation of the last two terms of (1) be zero, i.e., Fb _ DCS _ y = − N sin θ cos φ − 2 N e sin θ ul (33) Fb _ DCS _ z = − N sin θ sin φ − 2 N e sin θ lr . Substituting them into (2) yields Tb _ prop _ y = N (lTVC + l DCS ) sin θ sin φ + 2 N e ( lTVC + l DCS ) sin θ lr Tb _ prop _ z = − N ( lTVC + l DCS ) sin θ cos φ (34) + 2 N e ( lTVC + l DCS ) sin θ ul . Thus, the moving angle of the main TVC system in normal condition (we omit the last term in the right hand side of (34)) can be calculated as follows: Tby2 + Tbz2 Tby sin θ = , sin φ = . (35) N (lTVC + l DCS ) Tby2 + Tbz2 When the command torque is greater than the torque that the main TVC system can provide, we turn on the four extra TVC mechanisms. Among the four extra TVC systems, the upper and the lower extra TVC mechanisms only generate the torque in the z direction. Similarly, the left and the right ones only generate the toque in the y direction. Thus, we have: Tleft + Tright = 2 N e lTVC sin θ lr (36) Tupper + Tlower = − 2 N e lTVC sin θ ul . Therefore, the loss torque Tneed _ y and Tneed _ z (pitch and yaw) can be derived as: Tneed _ y = Tb _ y − Tb _ mainTVC (θ =15D ) _ y = 2 N e lTVC sin θ lr Tneed _ z = Tb _ z − Tb _ mainTVC (θ =15D ) _ z = − 2 N e lTVC sin θ ul . (37) As a consequence of this, −1 _y θ lr = sin −1 2Tneed N e lTVC , θ ul = sin

Tneed _ z − 2 N e lTVC

.

(38)

V. SIMULATION Two proposed missile’s integrated G/A systems are compared by simulation in this section. For using the simple notation, the missile’s integrated G/A system with SMC GL

is called A, while with DSMC GL is called B. the missile systems are examined for high-tier range, i.e., the missiles travel through the exo- and endo-atmosphere (a best way to test the feasibility and performance of the whole-phase-single-strategy controller). The initial position and velocity of the target and the missile are listed in Table 1. The target has horizontal velocity of 5 Mach moving in 270 km far and 158.76 km high with free-fall motion, whereas the defense missile is settled 10 km in the north of a city and the launch velocity is 14 m/s. The simplified aerodynamic model from [1] is adopted. The intercepting time of A is 110.12 sec, while that of B is 106.52 sec. Figure 5(a) ~ (g) is for system A, and Figure 5(h) ~ (n) is for system B. In Figure 5(a)(h), the chattering phenomenon is alleviated in B, whereas that is conspicuous in A although we use saturation function rather than sign function. In other words, DSMC GL is superior to SMC GL. In Figure 5(b)(i) and Fig 5(c)(j), the DCS thrust and thrust torque in A vibrate frequently, while that in B is relatively level. That is to say, the missile in C may fly with a long distance and therefore it can intercept the farer target. Furthermore, for DCS thrust, the unpleasant extra force generated by TVC is quite smooth in B, which leads to more stability of the airframe. Likewise, that A needs large thrust torque is neither efficient nor practical during flying. In Figure 5(d)(k), the aerodynamics produces huge effect inside atmosphere, while affects little outside it in the final intercepting phase. For TVC systems (Figure 5(e-g)(l-n)), the main TVC angle in A often reaches the constraint 15 degrees in the endoatmosphere. Therefore, we need to ignite the extra TCV mechanisms many times. Every time we ignite the extra TVC mechanisms, the airframe may be unsteady and then be crashed. Moreover, the main TVC system oscillates persistently with large angle for 40 seconds in A, which may cause the instability of motion of both TVC and the airframe during the fly course. In summary, all plots in B are much smoother than those in A, and B is more practical in real world. So, the performance of B is much more excellent than that of A though all these systems achieve the same goal (intercept the target successfully). In other words, the DSMC is better than the SMC and it can successfully avoid (or reduce) the chattering problem. Furthermore, the interception time in B is faster than that in A. Besides, these systems have verified our missile’s integrated G/A system designs. VI. CONCLUSION The decision and control part in the missile: GL and autopilot are discussed. We advocate a wingless airframe to reduce the effect from the aerodynamics as much as possible, and propose a robust autopilot system based on multiple TVC systems and DCS. The multiple TVC systems can be divided into two parts: the main TVC mechanism and the four extra TVC mechanisms that are the auxiliary systems to assist the main TVC when the main TVC system reaches the limitation

5252

FrB10.5 International Conference on Control Applications, vol. 1, 2004, pp. 812-818. [18] K. Ki-Seok and K. Youdan, "Robust backstepping control for slew maneuver using nonlinear tracking function," IEEE Transactions on Control Systems Technology, vol. 11, pp. 822-829, 2003. (a) Guidance Command 15

(h) Guidance Command

20 15

10

10

5

(m/s2)

5

(m/s2)

of the moving angle. Moreover, two missile guidance laws are derived, and Lyapunov stability analysis is used to guarantee the stability of the equilibrium point. The DSMC GL eliminates the chattering phenomenon successfully and holds the high aiming accuracy in the interceptions with excellent performance. In addition, we propose a new autopilot controller using backstepping control technique. Finally, various simulations are demonstrated to verify the feasibility of the integrated G/A systems of missiles. Table 1 Simulation Scenario

0

0

-5

-5

-10

aM⊥1 -10

aM⊥2

aM⊥1 aM⊥2

-15

aM⊥3

Y 135,000 10,000

Z 158,760 0

0

20

40

60

80

100

aM⊥3 -20

120

0

20

40

60

80

100

120

(i) DCS Thrust

(b) DCS Thrust

6000

8000

FDy

FDy

6000

FDz

4000

FDz

4000

2000

2000

(Nt)

Target Missile

X -233,827 0

Initial Velocity (m/s) X Y Z 1,299 -750 0 0 -10 10

(Nt)

Initial Position (m)

-15

0

0 -2000

-2000 -4000

-4000

REFERENCES

-6000

-6000

0

20

40

60

80

100

120

-8000

(c) Thrust Torque

4

x 10

3

0

20

40

60

80

100

x 10

prop

prop

Tbx

Tbx

prop

Tby

prop

2

Tby

prop

prop

Tbz

1

120

(j) Thrust Torque

4

3

2

Tbz

(Nt ⋅ m)

(Nt ⋅ m)

1

0

0

-1

-1

-2

-2

-3 0

20

40

60

80

100

-3 0

120

(d) Aerodynamic Force

5

2

x 10

20

40

60

80

100

120

(k) Aerodynamic Force

4

6

1.5

x 10

4

1

2

(Nt)

(Nt)

0.5

0

0

-2

aero

-0.5

aero

Fbx

Fbx

aero

Fby

-1

aero

Fby

-4

aero Fbz

-1.5 0

20

40

60

80

100

aero

Fbz -6 0

120

10

(deg)

12

10

(deg)

60

80

100

120

100

120

100

120

100

120

14

12

8

8

6

6

4

4

2

2

0

20

40

60

80

100

0

120

(f) extra TVC moving angle θ ul

20

10

5

5

(deg)

15

10

0

-5

-10

-10

-15

-15

0

20

40

60

80

100

-20

120

(g) extra TVC moving angle θ lr

20

0

20

15

10

10

5

5

(deg)

60

80

40

60

80

(n) extra TVC moving angle θ lr

20

0

40

(m) extra TVC moving angle θul

15

0

-5

-5

-10

-10 -15

-15 -20

20

0

-5

-20

0

20

15

(deg)

40

(l) main TVC moving angle θ

16

14

0

20

(e) main TVC moving angle θ

16

(deg)

[1] P. Zarchan, Tactical and Strategic Missile Guidance, vol. 199, 4th ed: American Institute of Aeronautics and Astronautics, Inc, 2002. [2] E. Duflos, P. Penel, and P. Vanheeghe, "3D guidance law modeling," IEEE Transactions on Aerospace and Electronic Systems, vol. 35, pp. 72-83, 1999. [3] J. Shinar and T. Shima, "A game theoretical interceptor guidance law for ballistic missile defence," in Conference on Decision and Control, vol. 3. Kobe, Japan, 1996, pp. 2780-2785. [4] C. L. Lin, H. Z. Hung, Y. Y. Chen, and B. S. Chen, "Development of an integrated fuzzy-logic-based missile guidance law against high speed target," IEEE Transactions on Fuzzy Systems, vol. 12, pp. 157-169, 2004. [5] S. Eun-Jung and T. Min-Jea, "Three-dimensional midcourse guidance using neural networks for interception of ballistic targets," IEEE Transactions on Aerospace and Electronic Systems, vol. 38, pp. 404-414, 2002. [6] J. Moon, K. Kim, and Y. Kim, "Design of missile guidance law via variable structure control," Journal of Guidance Control and Dynamics, vol. 24, pp. 659-664, 2001. [7] Y. Pin-Jar, "Optimal guidance of proportional navigation," IEEE Transactions on Aerospace and Electronic Systems, vol. 33, pp. 1007-1012, 1997. [8] C. D. Yang and C. C. Yang, "Optimal pure proportional navigation for maneuvering targets," IEEE Transactions on Aerospace and Electronic Systems, vol. 33, pp. 949-957, 1997. [9] J. R. Wertz, Spacecraft Attitude Determination and Control: Kluwer Academic Publishers, 1978. [10] Y. Fu-Kuang, C. Hsiuan-Hau, and F. Li-Chen, "Nonlinear optimal sliding mode midcourse controller with thrust vector control," in American Control Conference, vol. 2. Anchorage, AK, 2002, pp. 1348-1353. [11] G. P. Sutton, Rocket Propulsion Elements, sixth ed: John Wiley & Sons, Inc., 1992. [12] I.-F. Lin, "Novel Control Design for Intercepting Missiles with Highly Maneuverable Autopilot System," in Electrical Engineering, vol. Master. Taipei: National Taiwan University, 2006. [13] C. Chia-Hao, Y. Fu-Kuang, and F. Li-Chen, "New guidance laws and autopilot designs for intercepting missiles with TVC and DCS," 2005. [14] Y. Fu-Kuang, C. Kai-Yuan, and F. Li-Chen, "Variable structure-based nonlinear missile guidance/autopilot design with highly maneuverable actuators," Control Systems Technology, IEEE Transactions on, vol. 12, pp. 944-949, 2004. [15] H. Sira-Ramírez, "On the dynamical sliding mode control of nonlinear systems," Internation Journal Control, vol. 57, No. 5, pp. 1039-1061, 1993. [16] H. K. Khalil, Nonlinear Systems, 3rd ed: Prentice-hall. Inc., 2000. [17] M. C. Mickle, R. Huang, and J. J. Zhu, "Unstable, nonminimum phase, nonlinear tracking by trajectory linearization control," in IEEE

-8000

0

20

40

A

60

80

100

t (sec)

120

-20

0

20

40

B

60

80

t (sec)

Fig. 5. Missile’s integrated system A (a-g) and B(h-n) 5253