Head Pursuit Optimal Adaptive Sliding Mode Guidance Law

13th IFAC Symposium on Large Scale Complex Systems: Theory and Applications July 7-10, 2013. Shanghai, China

Head Pursuit Optimal Adaptive Sliding Mode Guidance Law Kefan Xiao, Bo Sun, Weidong Zhang, Yunze Cai Department of Automation and Key Laboratory of System Control and Information Processing, Ministry of Education Shanghai Jiao Tong University, Shanghai, China [email protected] Abstract: A head pursuit optimal adaptive sliding mode guidance law is proposed to intercept high speed target in a novel head pursuit scenario with low maneuver requirement and energy consumption. Head pursuit is a new trajectory strategy which the interceptor with smaller velocity is placed in front of the target and both of them fly in the same direction. This engagement could reduce relative speed and energy consumption by its special trajectory strategy. Optimal strategy is derived on the basis of minimal control energy to achieve satisfactory precision. But this optimal strategy could only deal with non-maneuver target. Thus, an adaptive sliding mode method is implemented to eliminate the error, which is caused by the target maneuver. And implementation of this method does not need to know the detail information of target acceleration but some estimation to its limitation. What’s more, with the constraint the optimal strategy exerts on the acceleration, the maneuver requirement is reduced and the energy consumption is cut down. The robustness and stability of this method has been proved theoretically. At last, the performance and precision of this method is verified with some numerical instances. Keywords:

Head pursuit, optimal adaptive sliding mode, guidance law, target maneuver, energy consumption.

1. Introduction

by controlling both of angles of velocity vector relative to line-of-sight (LOS) to be zero. However, this method was

The interception of the ballistic missiles (BMs) with high

derived from ideal geometry so it could only work well in

speed is always a challenge area. Various guidance methods and

engagement

strategy

of

trajectory

have

ideal environment. It also require the interceptor to be high

been

maneuver for the acceleration climax of the interceptor

implemented on this area. A typical engagement is head-on.

could be as 3 times high as target maneuver.

But interceptor and target in this engagement have very high

The optimal control theory has been generally applied on

closing speed that is a challenge for onboard seekers since

the guidance area to improve the performance. It could

they need to obtain some critical information such as

achieve

velocity from large distance precisely and quickly.

precision

requirement

by

low

control

acceleration and energy. Most of their solutions had

Recently, a special terminal trajectory performing was

feedback

proposed in [1] that the intercepting missile is located in

forms

which

were

derived

from

the

linear-quadratic optimal control theory. Optimal guidance

front of the target and on its flight trajectory and both of

law could achieve satisfactory intercepting accuracy with

them fly in the same direction. In addition, the speed of the

some constraint on control signal. Optimal guidance laws

missile is designed to be slower than the target so that the

was proposed to intercept a non-maneuvering target but

target could close the interceptor from the tail of it. This

decelerated by the atmospheric drag in [3] and an

approach can greatly reduce the closing speed relative to the

accelerating target in [4]. These methods need the

head-on engagement. What’s more, the energy consumption

knowledge of many states which can be acquired by a

of this strategy is much economic compared with the

nonlinear state estimator that was designed in the passage. In

tail-chase engagement. Energy consumption is also a critical

[5] a close-form optimal guidance was proposed with the

factor in interception. In [2], a further method of Pure Head

consideration of the time-varying velocity. Although optimal

Pursuit was proposed to deal with this engagement. The pure

guidance could perform well in many circumstances, it has

head pursuit guidance law (PHPGL) achieves interception 978-3-902823-39-7/2013 © IFAC

the

some critical limitations. When dealing with highly 508

10.3182/20130708-3-CN-2036.00096

IFAC LSS 2013 July 7-10, 2013. Shanghai, China

maneuver target, the implement of the optimal guidance law

guidance law too.

need the information of the target what is hard to acquire or

2.

know for sure. Even small errors could lead to totally

Problem Formulation As stated above, the interceptor is located in front of the

different results in the optimal guidance. Thus we need

target in the Head Pursuit engagement. In terminal process,

some approach to deal with this uncertainty of control

the accelerations that exert on the target and interceptor

information and make the guidance system to be robust.

only change the direction of the speed vectors rather than

The sliding-mode control method provides systematic

change the magnitude of them. Thus, the directions of

way to deal with modeling imprecision as well as

accelerations of both missiles are perpendicular with the

disturbance coming from outside. In [6] and [7] the main

directions of velocity, respectively. In addition, if the

advantage of sliding-mode control method was proposed that the implement of this method could transfer the system to be insensitive to the disturbances and modeling errors. Although it has advantages such as robustness, pure sliding-mode control also presents some drawbacks like it could lead to large energy consumption or fluctuation at the end period. In [8], the sliding-mode control law had been implemented on an air-air interception scenario which is a

Vq2  + uR − ωR V= R

(1)

VV − R q + u q − ωq Vq = R

(2)

interceptor doesn’t rotate, the relative motion between

non-linear system. In [9], a method that integration of both

interceptor and missile could be decoupled into two

the optimal control and sliding-mode control was applied

independent as well as perpendicular movements. In this

on design guidance for homing-missile against target

paper, the vertical plane is chosen as an example. The

maneuvers. But this method simplified the disturbance to be

terminal geometry is shown in Fig.1 with the planar

a constant which could rarely appear. In [10], an adaptive

form. Velocity and maneuvering acceleration vectors are

guidance law was proposed which take the autopilot

represented by V, a, respectively. And the flight path angles

dynamics into consideration. But the model it deals with

are donated by 𝛼𝛼, 𝛽𝛽. The angle of Line-of-Sight (LOS)

has been largely simplified.

relative to fixed coordinates is donated to q. R is the relative

This paper mainly deals with intercepting high speed

distance.

target within the head pursuit engagement to achieve low

VI

interceptor maneuver and satisfactory accuracy by using the head pursuit optimal adaptive sliding mode guidance law

α

aI

(HPOASM). The maneuver requirement of this method

q

exert on the intercepting missile is low. Consequently, the

I

energy consumption of this method is relatively low. The method adopts different control strategy from the PHPGL

VT

aT

presented above. HPOASM accomplish interception task by leading the angle rate of LOS to be zero which could realize

T

target approach interceptor in parallel. By implement the

q

R

β

Figure 1: Planar Engagement Geometry.

optimal strategy could lead to relatively low and stable

Define the kinetic equations of the system as follow:

acceleration signal with satisfied intercepting accuracy. At the same time, an adaptive sliding mode control method is introduced to deal with the error caused by target maneuver. This HPOASMGL could reduce the climax acceleration of the interceptor and thus decrease the energy consumption.

VR = R = VI cos (α − q ) − VT cos( β − q )

(3)

Vq = Rq = VI sin ( α − q ) − VT sin( β − q )

(4)

Differentiate both of the equations result in:

The intercepting precision could be maintained by this 509

IFAC LSS 2013 July 7-10, 2013. Shanghai, China

VR = q  −VT sin ( β − q ) + VI sin (α − q )  +  −VT cos ( β − q ) + VT β sin ( β − q )  + VI cos (α − q ) − VI α sin (α − q ) 

guidance law we want to devise is leading the state 𝑥𝑥 to be zero to realize the paralyze approaching.

(5)

3. Optimal adaptive sliding guidance law In this section, we mainly derive the guidance law to achieve satisfactory intercepting precision and low energy consumption. The process is divided into two parts: obtaining optimal solution in an ideal scenario firstly and using the adaptive sliding mode method to eliminate the error caused by target maneuver secondly. The HPOASMGL based the derivation is given at the end of this section.

˙

= Vq q VT cos ( β − q ) − VI sin (α − q )   ˙  − VT sin ( β − q ) + VT β cos ( β − q )    ˙   + VI sin (α − q ) − VI α cos (α − q )   

(6)

And, assume that: ˙

ωR = − VT cos ( β − q ) + VT β sin ( β − q )

(7)

˙

= uR VI cos (α − q ) − VI α sin (α − q )

3.1 Optimal solution

(8)

Different with PHPGL, we achieve interception by

˙

= ωq VT sin ( β − q ) + VT β cos ( β − q )

(9)

leading the angle rate of LOS to be zero. In this circumstance, the target could approach interceptor in

˙

= uq VI sin (α − q ) − VI α cos (α − q )

(10)

parallel. As we presents above, the state equation is 𝑥𝑥̇ = 𝑎𝑎(𝑡𝑡)𝑥𝑥 + 𝑏𝑏(𝑡𝑡)𝑢𝑢𝑞𝑞 − 𝑏𝑏(𝑡𝑡)𝜔𝜔𝑞𝑞

As the Eq. (5)-(8) showing, it’s apparent that 𝜔𝜔𝑅𝑅 and 𝑢𝑢𝑅𝑅

While 𝑎𝑎(𝑡𝑡), 𝑏𝑏(𝑡𝑡) are the substitutions of counterparts in Eq.

respectively represent the component of acceleration of

(13a) as follow:

target and interceptor on the LOS. And 𝜔𝜔𝑞𝑞 and 𝑢𝑢𝑞𝑞 are

respectively the representation of component of

𝑎𝑎(𝑡𝑡) =

acceleration of target and interceptor that the direction is perpendicular with the LOS. Substitute the counterparts in

𝑏𝑏(𝑡𝑡) =

Eq. (3) and (4) with Eq. (5)-(8) and we could get:

1 𝑅𝑅(𝑡𝑡)

(15) (16)

has no maneuver. Consequently Eq.[14] becomes

acceleration of target is perpendicular with the velocity

𝑥𝑥̇ = 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑢𝑢𝑞𝑞

direction. Thus, 𝜔𝜔𝑞𝑞 could be represented as:

be selected as follow:

𝑡𝑡𝑓𝑓

𝐽𝐽�𝑢𝑢𝑞𝑞 � = 𝜎𝜎𝜎𝜎�𝑡𝑡𝑓𝑓 � + � 𝛿𝛿(𝑡𝑡) 𝑢𝑢𝑞𝑞2 𝑑𝑑𝑑𝑑

(10):

𝑡𝑡0

2𝑅𝑅̇ 1 1 (12) 𝑞𝑞̇ + 𝑢𝑢𝑞𝑞 − 𝜔𝜔𝑞𝑞 𝑅𝑅 𝑅𝑅 𝑅𝑅 The critical point of designing the guidance law is to

(18)

In this function, σ > 0 is constant and 𝛿𝛿(𝑡𝑡) > 0. The first

𝑞𝑞̈ = −

term in order to guarantee the accuracy of the guidance law, we assume σ → ∞ because x�𝑡𝑡𝑓𝑓 � → 0 when σ → ∞. 𝑡𝑡0

devise 𝑢𝑢𝑞𝑞 to lead the angle rate of LOS 𝑞𝑞̇ to be zero

and 𝑡𝑡𝑓𝑓 respectively represent the beginning and ending

which could guide the interceptor to be reached by the

time.

target from its tail. Thus, by assuming x = 𝑞𝑞̇ to be state vector and 𝑅𝑅, 𝑅𝑅̇ to be related with time t only, we could get

The input signal 𝑢𝑢𝑞𝑞 that minimizes J�𝑢𝑢𝑞𝑞 � has the form as

follow:

𝑢𝑢𝑞𝑞∗ = − 𝛿𝛿(𝑡𝑡)−1 𝑏𝑏(𝑡𝑡)𝑝𝑝(𝑡𝑡)𝑥𝑥

(13a)

And 𝑝𝑝(𝑡𝑡) could be calculated from the Riccati-equation 𝑝𝑝̇ (𝑡𝑡) = −2𝑎𝑎(𝑡𝑡)𝑝𝑝(𝑡𝑡) + 𝛿𝛿(𝑡𝑡)−1 𝑏𝑏(𝑡𝑡)2 𝑝𝑝(𝑡𝑡)2

(13b)

Set 𝛿𝛿(𝑡𝑡) = −

In Eq. (13a) 𝑢𝑢𝑞𝑞 is the input signal. And we perceive target

acceleration component 𝜔𝜔𝑞𝑞 as disturbance. The goal of

(17)

To achieve the outcome we need, the cost function should

𝜔𝜔𝑞𝑞 = 𝑎𝑎 𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐(𝛽𝛽 − 𝑞𝑞) (11) Note that 𝑉𝑉𝑅𝑅 = 𝑅𝑅̇ and 𝑉𝑉𝑞𝑞 = 𝑅𝑅𝑞𝑞̇ , we could get from Eq.

y=x

2𝑅𝑅̇(𝑡𝑡) 𝑅𝑅(𝑡𝑡)

We assume the disturbance 𝜔𝜔𝑞𝑞 to be zero, which the target

In the terminal process, as presented above, the

the state equations: 2𝑅𝑅̇(𝑡𝑡) 1 1 𝑥𝑥̇ = − 𝑥𝑥 + 𝑢𝑢𝑞𝑞 − 𝜔𝜔 𝑅𝑅(𝑡𝑡) 𝑅𝑅(𝑡𝑡) 𝑅𝑅(𝑡𝑡) 𝑞𝑞

(14)

510

1 𝑅𝑅̇

which 𝑅𝑅̇ < 0 so that we could obtain

𝛿𝛿(𝑡𝑡) > 0. Thus, we could get

(19) (20)

IFAC LSS 2013 July 7-10, 2013. Shanghai, China

𝑝𝑝(𝑡𝑡) =

3𝑅𝑅(𝑡𝑡)4 𝑅𝑅(𝑡𝑡)3 − 𝑅𝑅(𝑡𝑡𝑓𝑓 )3

Combine Eq. (19) with Eq. (18) and notice that 𝑏𝑏(𝑡𝑡) = the optimal solution could be acquired: 3R(t)3 Ṙ (t) u*q = x R(t)3 -R(tf )3

� represents the estimate error to the upper limit of 𝑀𝑀 � = 𝑀𝑀 − 𝑀𝑀 � . To obtain disturbance which define as 𝑀𝑀

(21) 1

stability, the derivation of V on time need to be lower than zero which means 𝑉𝑉̇ < 0.

,

𝑅𝑅(𝑡𝑡)

Differentiate Eq. (26) in time and take Eq. (25) and (24)

into consideration:

(22)

1 � 𝑀𝑀 �̇ 𝑉𝑉̇ = 𝑠𝑠�𝑎𝑎(𝑡𝑡)𝑠𝑠 + 𝑏𝑏(𝑡𝑡)𝑢𝑢𝑞𝑞∆ − 𝑏𝑏(𝑡𝑡)𝜔𝜔𝑞𝑞 � − 𝑀𝑀 𝜖𝜖

In this solution, 𝑅𝑅(𝑡𝑡𝑓𝑓 ) means the distance when the input

In this case, 𝑏𝑏(𝑡𝑡) is definitely nonzero. Thus define 𝑢𝑢𝑞𝑞∆

is terminated which is caused by burning out of fuel.

𝑢𝑢𝑞𝑞∆ = −𝑏𝑏(𝑡𝑡)−1 (𝑎𝑎(𝑡𝑡)𝑠𝑠 + 𝐾𝐾𝐾𝐾 + 𝜃𝜃(𝑡𝑡)𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑠𝑠))

3.2 Optimal adaptive sliding mode guidance

Substitution of 𝑢𝑢𝑞𝑞∆ in Eq. (27) by Eq. (28) could result:

non-maneuver to achieve low miss distance with relatively

1 � 𝑀𝑀 �̇ (29) 𝑉𝑉̇ ≤ −K𝑠𝑠 2 − 𝜃𝜃(𝑡𝑡)|𝑠𝑠| + |𝑠𝑠||𝑏𝑏(𝑡𝑡)|𝑀𝑀 − 𝑀𝑀 𝜖𝜖 � so that Eq. (29) could be transferred Make 𝜃𝜃(𝑡𝑡) = |𝑏𝑏(𝑡𝑡)|𝑀𝑀

low energy consumption. However, it can’t be used on intercept maneuver target because optimal control is sensitive to disturbance. Thus, we design the adaptive

as:

sliding mode to deal with the state error caused by

1 � − 𝑀𝑀 � 𝑀𝑀 �̇ 𝑉𝑉̇ ≤ −K𝑠𝑠 2 + |𝑠𝑠||𝑏𝑏(𝑡𝑡)|𝑀𝑀 𝜖𝜖

disturbance. At last, we combine both solutions to generate the optimal adaptive sliding mode guidance law for head

Here we could acquire the adaptive law as: �̇ = 𝑀𝑀 � = 𝜖𝜖|𝑠𝑠||𝑏𝑏(𝑡𝑡)| 𝑀𝑀

pursuit engagement. Eq. (13) is the state equation with disturbance that

we do not need to know the detail of disturbance but

(23)

estimation of it. The parameter could vary with the process of interception.

optimal solution 𝑢𝑢𝑞𝑞 ∗ . We could separate the 𝑢𝑢𝑞𝑞 into two parts: 𝑢𝑢𝑞𝑞 and

Therefore, the final 𝑢𝑢𝑞𝑞∆ could be acquired with the

Therefore, by minus Eq. (13) with Eq.

derivation above:

(21), we can obtain:

∆𝑥𝑥̇ = 𝑎𝑎(𝑡𝑡)∆𝑥𝑥 +

𝑏𝑏(𝑡𝑡)𝑢𝑢𝑞𝑞∆

− 𝑏𝑏(𝑡𝑡)𝜔𝜔𝑞𝑞

� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑠𝑠)) 𝑢𝑢𝑞𝑞∆ = −𝑏𝑏(𝑡𝑡)−1 (𝑎𝑎(𝑡𝑡)𝑠𝑠 + 𝐾𝐾𝐾𝐾 + |𝑏𝑏(𝑡𝑡)|𝑀𝑀 �̇ = 𝑀𝑀 � = 𝜖𝜖|𝑠𝑠||𝑏𝑏(𝑡𝑡)| 𝑀𝑀

(24)

∆𝑥𝑥 = 𝑥𝑥 − 𝑥𝑥 ∗ means state error and 𝑢𝑢𝑞𝑞∆ = 𝑢𝑢𝑞𝑞 − 𝑢𝑢𝑞𝑞 ∗ is the

disturbance to be zero and make the ideal situation happen.

we design the adaptive sliding mode method is to lead the

We could obtain the optimal adaptive sliding mode

error state ∆𝑥𝑥 to be zero which is caused by the

guidance law as follow:

disturbance 𝜔𝜔𝑞𝑞 .

𝑢𝑢𝑞𝑞 = 𝑢𝑢𝑞𝑞∆ + 𝑢𝑢𝑞𝑞 ∗ =

Firstly, we could acquire the estimation of the limitation

of disturbance:

Define the sliding surface:

s = ∆𝑥𝑥

3𝑅𝑅(𝑡𝑡)3 𝑅𝑅̇(𝑡𝑡) ∗ 𝑥𝑥 𝑅𝑅(𝑡𝑡)3 − 𝑅𝑅(𝑡𝑡𝑓𝑓 )3

(33)

− 𝑏𝑏(𝑡𝑡)−1 (𝑎𝑎(𝑡𝑡)∆𝑥𝑥 + 𝐾𝐾∆𝑥𝑥 � 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(∆𝑥𝑥)) + |𝑏𝑏(𝑡𝑡)|𝑀𝑀

Both 𝑥𝑥 ∗ and x have the same initial value which is the

(25)

angle rate of LOS at the beginning of the intercepting

The goal to design the approaching method is to drive the

process. Consequently, initial ∆𝑥𝑥 is 0. With this guidance

variable sliding on the s = 0 plant. Thus, we would derive

law, we do not need to know the detail of the target

the solution and prove its stability at the same time. Firstly,

acceleration but the evaluation of it. In addition, the

the Lyapunov function is proposed as follow: 1 1 2 � V = 𝑠𝑠 2 + 𝑀𝑀 2 2𝜖𝜖

(32)

This solution could drive the state error that is caused by the

acceleration of interceptor we need to design. So, the goal

0 < �𝜔𝜔𝑞𝑞 � < M, M is a positive constant.

(31)

Then 𝑉𝑉̇ ≤ −K𝑠𝑠 . This means 𝑉𝑉̇ ≤ 0 . Thus s → 0 and � → 0 could be achieved. Using this kind of adaptive law, 𝑀𝑀

The superscript * means that the state is contributed by 𝑢𝑢𝑞𝑞∆ .

(30)

2

happened in real situation. Then, represent the state

∗

(28)

𝐾𝐾 and 𝜃𝜃(𝑡𝑡) is adaptive parameter that need to devise latter.

The optimal solution could be used to intercept

equation without disturbance as follow: 𝑥𝑥 ∗̇ = 𝑎𝑎𝑥𝑥 ∗ + 𝑏𝑏𝑢𝑢𝑞𝑞 ∗

(27)

maneuver acceleration of interceptor is constrained by the optimal strategy so that is relatively low. Thus, the energy

(26)

consumption is relative low compared with most of other 511

IFAC LSS 2013 July 7-10, 2013. Shanghai, China

guidance and intercepting precision could be maintained at

which is the acceleration the interceptor must be exerted.

the same time.

Thus the function could estimate the energy consumption at some extent. In Fig.5, we could clear see that the guidance

4. Simulation

law we presented consume much lower energy than the

In this section, simulation of the head pursuit optimal

PHPGL does.

adaptive sliding-mode guidance law (HPOASM) and the comparison with the Pure Head Pursuit guidance law (PHPGL) are proposed. In this simulation, target and interceptor move with constant velocity 𝑉𝑉𝑇𝑇 = 1900 and 𝑉𝑉𝐼𝐼 = 1600 . The distance between them is 𝑟𝑟0 = 3000. And the initial flying

angle of both target and interceptor are 𝛼𝛼 = 𝛽𝛽 = 0° which

are big heading errors. The accelerations of target are

choose to be three different values which are 20g, 0, −20g. Design the parameters of the guidance as 𝐾𝐾 = − 𝜖𝜖 = 6

[11]

2𝑅𝑅̇0 𝑅𝑅

,

which 𝑅𝑅̇0 is the initial distance rate. According

to the formulas we derived above, the simulation could be done as follow.

Figure 2: Flying Trajectory in Fixed Inertial Coordinates.

In Fig. 2, three inertial trajectories of the interceptor and target which are exerted with three different accelerations are presented in fixed coordinates. It’s clear that this guidance law could achieve satisfactory precision in these high maneuver situations as well as non-maneuver circumstance. In Fig. 3, take the acceleration of target that is 20g as an example to show the angle rate of LOS. We could see that the angle rate of LOS is driven to zero by this guidance law even the target is highly maneuver. This means the target and interceptor could approach each other parallel. In Fig. 4, the control accelerations of HPOASM and PHPGL are presented respectively. It’s apparent that the acceleration of HPOASM is more stable than PHPGL due

Figure 3: Angle rate of LOS.

to the constraint of optimal strategy. The climax of the acceleration is also significantly lower than the PHPGL does. Thus, the requirement of maneuver of HPOASM is lower than PHPGL. It also could be seen that the curve of acceleration that is derived from the HPOASM is relatively smooth. It’s because that we use the function of saturation, which could largely reduce the chattering, in the simulations. 𝑡𝑡

Last, we choose the function 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸(𝑢𝑢, 𝑡𝑡) = ∫𝑡𝑡 𝑢𝑢2 𝑑𝑑𝑑𝑑 0

to measure the energy consumption of the interceptor in the intercepting process. In the function, u is the control signal,

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IFAC LSS 2013 July 7-10, 2013. Shanghai, China

combination of them could deal with this intercepting problem well. This new guidance law which we derived could lower maneuver acceleration and consequently cut down energy consumption and maintain the performance as well as precision of interception.

Acknowledgement This paper is partly supported by the National Science Foundation of China under Grant 61025016, 11072144, 61034008, and 61221003.

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