Abstract— A new longitudinal trajectory tracking law for the Terminal Area Energy Management (TAEM) phase of the Reusable Launch Vehicle (RLV) is presented in this paper. The conventional PID method controls the height and the velocity by the angle of attack and the angle of airbrake respectively, and no coupling is considered. To improve the tracking precision, the new tracking law is designed based on the linear quadratic regulator (LQR) theory where the coupling is taken into account. Finally, the trajectory tracking law based on the LQR is simulated and compared with the conventional method. Simulation results indicate the effectiveness and the robustness of the new tracking law.

I. INTRODUCTION

Reusable Launch Vehicle (RLV) is a type of reusable aircrafts that can make a round trip between the earth and the outer space [1]. It is forecasted that RLV is the trend of space technology in the following years [2]. The reentry flight of RLV is consisted of three phases: initial entry phase, Terminal Area Energy Management (TAEM) phase and landing phase [3]. The TAEM phase starts from the point with a height of 30km and a velocity of 2.5Ma, and ends at the point with a height of 4km and a velocity of 0.5Ma. In the TAEM phase, the task is to dissipate surplus energy and adjust the heading direction towards the runway, so the RLV will land safely with a suitable velocity and height [4], which makes this phase a vital phase of the reentry flight. However, a large variation range of dynamic pressure and velocity, significant changes of aerodynamic characteristics along with the uncertain states of the RLV after initial entry phase also make the task of the TAEM phase a highly difficult work.

The general method of energy management in the TAEM phase is to divide the motion into the longitudinal motion and the lateral motion, then design the standard trajectory and corresponding reference guidance commands. In actual flight, guidance commands are adjusted to track the standard trajectory. As for longitudinal motion, two variables are to be controlled: height and velocity. Assuming these two variables are independent of each other, angle of attack is used to control the height and angle of airbrake is used to control the velocity of the RLV respectively. But actually, height and velocity are coupled and the separate design causes inaccuracies in the process of tracing the reference trajectory.

This paper aimes to improve the accuracy of height and velocity together by means of trajectory tracking law based on the linear quadratic regulator (LQR) theory. Firstly, the state

* Research supported by the National Nature Science Foundation of China (No.91116002, No.91216034, No.61333011 and No.61121003).

B. Zheng, Z, Liang, Q. Li, Z. Ren are with the Science and Technology on Aircraft Control Laboratory, Beihang University, Beijing, 100191 China (phone: 86-010-82314573-11; fax: 86-010-82313265; e-mail: changfeng@asee.buaa.edu.cn; muziqingdong@126.com)

equation of RLV in TAEM phase is to be calculated out. For acquiring feedback control, performance index is also computed. Finally the feedback matrix is obtained by resolving the Riccati equation. Input of the system is the feedback of the state.

II. TRAJECTORY TRACKING METHODS

Assuming that the earth is a flat surface and the RLV is a particle, the motion equation of RLV is as follows [5]:

- - sincos cos

sincos

sin

mv D mgmv L mg

Lmv

h v

γγ φ γ

φψγ

γ

=�� = −��� =��� =�

��

�

�

(1)

where m and v represents the mass and velocity of RLV; L and D represents the lift force and resistance force acting on RLV; �, �, � represents flight-path angle, velocity azimuth angle and bank angle of RLV respectively. Assuming that bank angle equals to zero, we can obtain the simplified motion equation of RLV is as follows:

sincos

sin

mv D mgmv L mg

h v

γγ γ

γ

� = − −� = − −�� =�

���

(2)

In order that RLV can land safely with a suitable velocity and height onto the runway, the standard trajectory and corresponding reference guidance commands are calculated in advance. In flight procedure, deviations of actual height and velocity from reference values are obtained, and then angle of attack and angle of airbrake are adjusted from reference values to eliminate these deviations [6]. In a common method, angle of attack is adjusted according to the deviation of height while angle of airbrake is adjusted according to the deviation of velocity. The following equations use PID methods as an example[6]:

1* 2* 3* *d hk h k k h dtdt

α ΔΔ = Δ + + Δ� (3)

4* 5* 6* *d vb k v k k v dtdt

δ ΔΔ = Δ + + Δ� (4)

where �� represents the difference between actual angle of attack and reference value; ��b represents the difference between actual angle of airbrake and reference value; �h represents the deviation of height; �v represents the deviation

Trajectory Tracking for RLV Terminal Area Energy Management Phase based on LQR

Bowen Zheng, Zixuan Liang, Qingdong Li, Zhang Ren

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Proceedings of 2014 IEEE Chinese Guidance, Navigation and Control Conference August 8-10, 2014 Yantai, China

978-1-4799-4699-0/14/$31.00©2014 IEEE

of velocity; k1, k2, k3 represent proportional gain, derivative gain, integral gain of deviation of height; k4, k5, k6 represent proportional gain, derivative gain, integral gain of deviation of velocity. Airbrake cannot be used when the velocity is too high to avoid damage to it. In most cases, airbrake is open when Mach number is under 0.8, so �b equals to zero when velocity is higher than 0.8Ma.

III. TRAJECTORY TRACKING LAW BASED ON LQR

As coupling exists in the height and velocity tracking laws in (8) and (9), the LQR method is used when both the angle of attack and the angle of airbrake are adjusted.

Before using trajectory tracking law based on LQR, the state equation of RLV in TAEM phase has to be calculated out. For longitudinal motion, height, velocity, flight-path angle are selected as state variables, angle of attack, angle of airbrake are selected as input variables and height, velocity are selected as output variables. Using means of small perturbation linearization, the state equation of RLV in TAEM phase are obtained as follows [7]:

h hv A v B

bαδ

γ γ

� �Δ Δ� �Δ� �� � � �Δ = Δ + � �� � � � Δ� �� � � �Δ Δ� �� �

�

��

(5)

where

2

0 sin cossin cos

cos cos sinh v

h v

vA D g D g

L g vL L g gv vv

γ γγ γ

γ γ γ

� �� �� �

= − − − −� �� �− − +� �� �

(6)

0 0

b

b

D DBL Lv v

α δ

α δ

� �� �− −� �=� �� �� �

(7)

�h, �v, �� represent errors of height, velocity, flight-path angle of RLV; ��, ��b represent increments of angle of attack, angle of airbrake on reference values; g represents gravitational acceleration; Lh, Lv, L�, L�b represent derivatives of lift force on height, velocity, angle of attack, angle of airbrake and given by (8); Dh, Dv, D�, D�b represent derivatives of resistance force on height, velocity, angle of attack, angle of airbrake and given by (9); v represents the velocity of RLV; L and D represents the lift force and resistance force acting on RLV.

, , ,h v bL L LL L L Lh v b

Lα δα δ∂ ∂ ∂ ∂= = = =∂ ∂ ∂ ∂

(8)

, , ,h v bD D DD D D Dh v b

Dα δα δ∂ ∂ ∂ ∂= = = =∂ ∂ ∂ ∂

(9)

Considering lowering the values of state variables and the input, performance index of the system is defined as [8]:

0

1 ( )2

T TJ x Qx U RU dt∞

= +� (10)

where X=[ h v �]T, U=[ � �b]T, Q and R are weighting matrix of deviation of state variables and input variables. According to ‘Bryson’s rule’ they are constant symmetric positive definite matrix, defining as follows:

2max

2max

2max

1 0 0

10 0

10 0

h

Qv

γ

� �� �� �� �

= � �Δ� �

� �� �

� �� �

(11)

2

max

2max

1 0

10R

b

α

δ

� �� �� �=� �� �� �

(12)

where �hmax, �vmax and ��max represent maximum allowable deviations of height, velocity and flight-path angle; ��max and ��bmax represent maximum rate of change of angle of attack and angle of airbrake. In order to obtain the minimum value of the performance index J defined in (10), theory of optimal linear regulator is used which provides the feedback matrix by resolving the Riccati equation:

1 0T TKA A K KBR B K Q−− − + − = (13)

Form of feedback control is obtained as follows:

h

K vb

αδ

γ

Δ� �Δ� � � �= Δ� � � �Δ� � � �Δ� �

(14)

where K represents the state feedback matrix of the system. The complete guidance commands are the sum of increments and reference values of angle of attack and angle of airbrake:

ref

refbb bαα αδδ δ

� �� � � �= +� �� � � �� � � �� � (15)

where � and �b represent complete guidance commands, �ref and �bref represent reference commands for angle of attack and angle of airbrake.

IV. SIMULATION

To test the proposed trajectory tracking law, a RLV model is used in simulations. The initial and terminal conditions for the TAEM phase of a RLV are as follows:

Initial velocity: 760m/s

Initial height: 30000m

Initial flight-path angle: -5°

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Remaining range: 100.66km

Terminal height: 4000m

Terminal velocity: 157m/s

A. Abbreviations and Acronyms Assuming that there is no dispersion, the state variables

height and velocity and the input variables angle of attack and airbrake are shown in Figs. 1-3:

0 20 40 60 80 100 1200

5

10

15

20

25

30

Range(km)

Hei

ght(

km)

reference trajectory

actual trajectory

Figure 1. Reference and actual height trajectories without dispersion

0 20 40 60 80 100 120100

200

300

400

500

600

700

800

Range(km)

Vel

ocity

(m/s

)

reference trajectory

actual trajectory

Figure 2. Reference and actual velocity trajectories without dispersion

0 20 40 60 80 100 120-5

0

5

10

15

20

25

Range(km)

Ang

le o

f at

tack

or

airb

rake

(deg

)

angle of attack

angle of airbrake

Figure 3. Angle of attack and airbrake without dispersion

Figs.1 and 2 display the reference height and velocity designed in longitudinal plan and actual height and velocity generated in actual flight. The two figures illustrate that the accuracy is sufficient when there is no dispersion. In Fig2, the velocity is dispelled late in flight because the airbrake is open only when the velocity is below 0.8Ma.

B. Dispersed cases When dispersions are considered in the TAEM phase,

effects of PID method and trajectory tracking law based on LQR are compared. To display this issue clearly, the dispersion is sorted into two categories: initial condition dispersion and aerodynamic parameter dispersion. Adding fluctuation on each item and the results of the two methods are calculated.

Initial condition dispersion includes initial error of height, velocity and flight-path angle. Aerodynamic parameter dispersion includes error of atmosphere density, lift coefficient and drag coefficient.

TABLE I. CASE DISPERSION VALUE

Case Dispersion items Dispersion value

1 Initial height +3km

2 Initial height -3km

3 Initial velocity +120m/s

4 Initial velocity -120m/s

5 Initial flight-path angle +5

6 Initial flight-path angle -5

7 Range +20%

8 Range -20%

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TABLE II. CASE DISPERSION VALUE

Case Dispersion items Dispersion value

1 Atmosphere density +30%

2 Atmosphere density -30%

3 Lift coefficient +20%

4 Lift coefficient -20%

5 Drag coefficient +20%

6 Drag coefficient -20%

7 Mass +5%

8 Mass -5%

0 20 40 60 80 100 1200

5

10

15

20

25

30

35

Range(km)

Hei

ght(

km)

Figure 4. Height trajectories for LQR in dispersed cases

0 20 40 60 80 100 120100

200

300

400

500

600

700

800

900

Range(km)

Vel

ocity

(m/s

)

Figure 5. Velctity trajectories for LQR in dispersed cases

-20 -15 -10 -5 0 5 10 15 20-500

-400

-300

-200

-100

0

100

200

300

400

500

Velocity(m/s)

Hei

ght(

m)

LQR method

PID method

Figure 6. Height and velocity error of PID and LQR method

Figs.4-6 illustrate that the accuracy of height and velocity under dispersion of initial condition are sufficient with means of trajectory tracking law based on LQR. The accuracy is promoted compared with PID method which does not consider the copula between height and velocity.

V. CONCLUSION The theoretical derivation and simulation results above

illustrate that the copula between height and velocity shall be taken into consideration to improve the accuracy of longitudinal trajectory tracking. Means of trajectory tracking law based on LQR is authenticated to be an effective method to fufill this target, especially under dispersion of aerodynamic parameter.

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[3] Thomas E. Moore, Lyndon B. Johnson Space Center, Houston, Texas, “Space Shuttle Entry Terminal Area Energy Management,” NASATM, vol. 104744, pp. 1-46, 1991.

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[8] Greg A. Dukeman, “Profile-following entry guidance using linear

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